claims depot shelf
The Collatz Conjecture
Formal claims (Lean)
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Starting from Nicomachus' original odd-number pattern, the paper's central discovery is that the cubic case is exceptional in two complementary ways. Locally, requiring $n^p=b^2-a^2$ with $b-a=n$ forces $a=(n^{p-1}-n)/2$ and $b=(n^{p-1}+n)/2$; these are exactly the consecutive triangular numbers $T_{n-1}$ and $T_n$ when $p=3$, since $b=T_n$ reduces to $n^{p-1}=n^2$. No other $p$ satisfies this for
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The central discovery is a theorem: for p a positive integer, any sequence satisfying a_{n+1}u_{n+1}(x)=b_n(x+d_n)u_n(x)-c a_n u_{n-1}(x) with a_0=0 and c\neq 0 obeys \sum_{n=0}^{p-1} \frac{b_n}{c^n}u_n(x)u_n(y)=\frac{a_p}{$c^{{p-1}}$(x-y)}\big(u_p(x)u_{p-1}(y)-u_{p-1}(x)u_p(y)\big), with the x=y limit giving a derivative identity. The paper also derives a companion identity with alternating signs
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On the paper's own terms, the discovery is that degenerate special numbers are governed by a pair of operator identities. The first expands $\left(x^{1-\lambda}\frac{d}{dx}\right)^n$ as $\sum_{l=0}^n \binom{n}{l}_{\lambda}\, x^{l-n\lambda}\left(\frac{d}{dx}\right)^l$, and the second rewrites $x^{n\lambda}\left(x^{1-\lambda}\frac{d}{dx}\right)^n$ as the $\lambda$-deformed Euler operator $\left(x\fr
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The paper's central claim is Theorem 1: there is a constant $c>0$ such that $|A\cap[1,x]|\geq cx$ for all sufficiently large $x$. The proof does not try to list $A$ directly. It encodes compositions of the maps $T_a(x)=ax-1$ as paths through nineteen half-open intervals, switches among four assignments of the multipliers $2,3,5,9,14$ to keep a three-dimensional prime-exponent imbalance near the or
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The paper proves a dichotomy for square exponents: when $k=2$, the generalized equation (1.2) has integer solutions only when the left side is a sum of two or of four consecutive arithmetic-progression terms. In both cases there are infinitely many solutions, and every solution is 'good,' meaning the quotient $\lfloor (m-b)/q \rfloor$ does not change when $m-b$ is replaced by $m-b-1$; geometricall
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The central discovery is Proposition 13: when f(x) is an even power series, the square of the Sprugnoli array M(g,f) equals ((g(x)g(-x)+$x^{2}$ f(x)^2))/(1-$x^{2}$)^2, which is an aerated element of the Appell subgroup of the Riordan group. Consequently, every aerated Appell array that can be written in this form has a square root that is a Sprugnoli array M(g,f). If the right-hand side is the con
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The central claim is Theorem 1.4: for $X$ and $Y$ independent of $q$, the unitizing operator of degree $3$ applied to $\frac{f_6}{f_2}(qX,q/X,qY,q/Y;q^2)_\infty$ returns its cube. The two auxiliary variables are not decorative; choosing $X$ and $Y$ among the fifth roots of unity $\rho,\rho^2,\rho^3,\rho^4$ produces the last pair of 3-convolutive products in Table 2, while the choices $1,-1,\omega,
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The central claim is that the rational function (1+cxt)/(1+axt+bt^2) generates a genuinely unified family: K_n(x|-2,1,-1)=T_n(x), K_n(x|-2,1,0)=U_n(x), K_n(x|-2,1,-1/x)=V_n(x), K_n(x|-2,1,1/x)=W_n(x). The paper establishes the recurrence K_n = -ax K_{n-1} - b K_{n-2} with K_0=1 and K_1=(c-a)x, an n×n tridiagonal determinant formula, and a closed form as a sum of binomial coefficients. It further p
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The main result, Theorem 3.25, states that for all n≥k≥1, kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k). This answers affirmatively the 1985 question of whether the second-kind Comtet numbers admit a recurrence involving only a fixed number of terms. The proof derives from two elementary identities satisfied by the weighted Stirling polynomial G_{n,k}(x): the polynomial recurrence G_{n+1,k}(x)=(x+k)G_
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The central claim is Corollary 5.3: for every integer k ≥ 0, Φ(z,s,a) = (8/3)H_k - 2H_{k+1} + (1/3)H_{k+2}, DΦ(z,s,a) = (-4H_k + 5H_{k+1} - H_{k+2})/(3·2^k), and D^2Φ(z,s,a) = (2H_k - 3H_{k+1} + H_{k+2})/(9·4^k). These follow from the operator identity H_k = (α_k D^2 + β_k D + 1)Φ, with α_k = (3/2)4^k and β_k = (3/2)2^k, combined with a Vandermonde inversion on the geometric factors 1, 2, 4. The p
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For every even positive integer h and every B_h-set A, the liminf as n tends to infinity of A(n) divided by the h-th root of n/log n is at most (pi/log 2 times Gamma(1+h/2)^2 over Gamma(1+1/h)^h) raised to the power 1/h.
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The paper establishes that for every N≥1 and 0≤r≤N, the maximum of C_min(d) over binary words of length N with exactly r ones is attained exactly, up to rotation, by the Christoffel word d^chr_{N,r}. The functional C(d) is defined from the parity word as C(d)=Σ 2^{i−1}3^{r_i(d)} d_i, where r_i(d) counts ones strictly to the right of position i; this is the numerator in the standard affine formula
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Framing every integer as a Mersenne word — digits 0,1,2 over weights 2^k − 1, with a 2 forcing all lower digits to 0 — yields two results at once. A successor transformation on the nonbinary sublanguage enumerates A055938 and gives the difference rule y_{n+1} − y_n = 3 − 2χ(n); this proves the conjectural identity A080578(n) = A055938(n−1) + 2 for n ≥ 2, and shows the binary-side counting function
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Framing every integer as a Mersenne word — digits 0,1,2 over weights 2^k − 1, with a 2 forcing all lower digits to 0 — yields two results at once. A successor transformation on the nonbinary sublanguage enumerates A055938 and gives the difference rule y_{n+1} − y_n = 3 − 2χ(n); this proves the conjectural identity A080578(n) = A055938(n−1) + 2 for n ≥ 2, and shows the binary-side counting function
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On its own terms, the paper proves Theorem 1: if an H-invariant Legendre pair of length 333 exists, then H is a subgroup of the kernel of the reduction (Z/333Z)^× → (Z/3Z)^× and |H| ≤ 6. Equivalently, inside the order-108 kernel there are exactly 30 subgroups; 21 are proved impossible, including every subgroup of order at least 9. The decisive analytic step treats the final order-9 subgroup H12 =
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On its own terms, the paper proves Theorem 1: if an H-invariant Legendre pair of length 333 exists, then H is a subgroup of the kernel of the reduction (Z/333Z)^× → (Z/3Z)^× and |H| ≤ 6. Equivalently, inside the order-108 kernel there are exactly 30 subgroups; 21 are proved impossible, including every subgroup of order at least 9. The decisive analytic step treats the final order-9 subgroup H12 =
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The central claim is that the new polynomials M_n(x)=n!∑ binom(n+k+1,n-k)x^k and N_n(x)=n!∑ binom(n+k,n-k)x^k, once rescaled to K_n, J_n, L_n, turn three Bell-polynomial binomial sums into finite polynomial sums (Theorems 3.4, 3.6, 3.7). The paper also derives their exponential generating functions, constructs Euler-Seidel matrices from them, and defines degenerate λ-versions K_{n,λ}, J_{n,λ}, L_{
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On its own terms, the central claim is Theorem 4: under the hypotheses p_i(0)≠0, p2 coprime to p1 and p3, and d2 ≤ d1+1, the k-th column (k≥1) of the Riordan array (p1/p2, t p3) over F_q is ultimately periodic with least period π, the least period of the coefficient sequence of 1/p2; its periodic block is C_k = A^k C_0, with A a π×π circulant matrix generated by the coefficients of p3. Theorem 8 e
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For every irrational α greater than (3+√5)/2 and every positive integer n there exists a necessary and sufficient elementary-number-theoretic condition that decides whether a natural number x belongs to the nth iterate of the Beatty sequence of modulus α.
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For every integer k≥1 and every n≥0 the Laurent coefficients J_{k,n}(m) of the normalized Jacobi tail are positive for all integers m with -(k+n)≤m≤k+n. Equivalently, the generating function of the tail equals z^{-k}(1+z+⋯+z^{2k})/((1-qz)(1-z^{-1}q))+z^{-k}N_k(z,q), where N_k has exclusively non-negative coefficients.
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The generalized q-Morgan-Voyce polynomials defined by M_n(x,q)=(x+1+q)M_{n-1}(x,q)-q^{n-2}M_{n-2}(x,q) admit generating functions and explicit formulas obtained via the Fibonacci operator and the binomial theorem, possess natural negative-index extensions, satisfy summation and determinantal identities, and obey a matrix-based q-Cassini identity.
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Every infinite exponent code generated by the accelerated Collatz orbit of a fixed positive odd integer forces both the 2-adic start residue rate and the 3-adic endpoint residue rate to vanish asymptotically. Near-critical codes produced by random sampling, mechanical construction, and adaptive evolutionary search at lengths 100–400 retain clearly positive rates on both residues, so they remain in
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For every integer k≥3 the sequence of kth powers is not maximal as a GD sequence; the sequences of integers congruent to 1 modulo k are maximal precisely when k=2 or 3 and are non-maximal for all k≥4; and the sequences of the form k^n-1 are non-maximal for every k≥3.
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There is a uniform recursion (Theorem 4.2) that expresses every Hankel determinant H_w(n) of a weighted binary sum-of-digits sequence in terms of two smaller determinants of the same type (or of the first-difference type). For ordinary sum-of-digits the recursion collapses on the indices n_k=⌈2^{k+2}/3⌉ to the elementary closed form (-1)^{(k+2)(k+3)/2}(k+1), giving an infinite explicit family.
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U(n) is odd if and only if n belongs to the even-offset class J, and even if and only if n belongs to the odd-offset class K; the successive elements of these classes are given by the closed forms j(n)=2n-⌊(√(8n+1)-1)/2⌋ and k(n)=2n+1+⌈(√(8n+9)-1)/2⌉. Consequently the parity of every Redheffer matrix is completely determined by evaluating one of these two formulas.
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For a broad class of classical sequences the dilated Hankel determinant ¨H_n(a)=det(a_{2i+j}) admits a simple product evaluation, even though no universal continued-fraction formula exists for it and the class of sequences with known product formulas is larger for the ordinary Hankel determinant. The evaluations are obtained by six methods developed in the paper and include, among others, the fact
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In any modular g-Golomb ruler the cyclic gaps themselves obey the multiplicity bound of g. Consequently the largest gap is at least the ceiling of (N + ga(a−1)/2 + ra)/n, where n = ag + r. Cutting at that gap yields an ordinary g-Golomb ruler whose diameter is strictly smaller than the diameter obtained from the classical average-gap cut. The same extraction applies after a controlled folding that
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In any modular g-Golomb ruler the cyclic gaps themselves obey the multiplicity bound of g. Consequently the largest gap is at least the ceiling of (N + ga(a−1)/2 + ra)/n, where n = ag + r. Cutting at that gap yields an ordinary g-Golomb ruler whose diameter is strictly smaller than the diameter obtained from the classical average-gap cut. The same extraction applies after a controlled folding that
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When the six GKP parameters are treated as indeterminates, each row sequence (T(n,k))_k is coefficientwise strongly log-concave, and the sequence of row-generating polynomials (P_n(x))_n is coefficientwise strongly log-convex (hence Hankel-totally positive of order 2) jointly in x and the six parameters.
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Lemma 2.1 asserts that for a nonnegative integer n and complex z outside {0,…,n}, ∑_{k=0}^n (z choose k)(-1)^k (x+1)^k equals (-1)^n (n+1)(z choose n+1) times ∑_{k=0}^n (n choose k) x^k/(z-k). Every later identity in the paper is obtained from this equality by substituting particular values, replacing z by half-integers, integrating with respect to x, or differentiating with respect to z.
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There exists an explicit constant γ0 = 0.943492590... < 0.9435 such that the largest Sidon subset of {0,...,N-1} satisfies F(N) ≤ N^{1/2} + γ0 N^{1/4} + O(1). The constant is realized by a concrete eight-kernel rational certificate that produces a product ab strictly less than (0.9435)^2 inside the vector smoothing lemma.
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For consecutive fixed points of J4 at least 3, the pure high extremal points between them form a finite alternating sequence of type-1 and type-2 blocks whose lengths, last points, and exit to the next fixed point are completely determined by the ternarity of the starting fixed point, the valuations u3(4np+2) or u3(4np+3), the transition meters u3(4ne+3) and u3(4ne+5), and an explicit termination
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The central claim is that the implicit equation a(Σ_{j=0}^{r−1}a(n−j))=n, for a nondecreasing positive-integer sequence a, has, besides the greedy solution, an explicit inhomogeneous Beatty solution a(n)=⌊n/√r+√r/2⌋ for every r≥2 that is not an even perfect square, and that this formula solves the equation for all n≥r. When r=(2m)², the same formula gives n+1 instead of n, and this failure is show
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The central claim is that the sequence (A_{d+2})_{d≥1} is sub-Fibonacci: A_3 = A_4 = 1 and A_{d+2} ≤ A_{d+1} + A_d for d > 2, with the stronger monotonicity A_d ≤ A_{d+1} holding for d > 2. Here A_d counts the finite O-sequences of multiplicity d whose last non-zero value is at least 2. The proof uses the identity A_d = 1 + Σ O(d−ℓ−1, d−1, i, d−j)·O(d−ℓ, i−1, k−1, j) over appropriate ranges, and s
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If a triangular number belongs to a D(a)-pair of triangular numbers, then it can be extended to infinitely many distinct D(a)-triples consisting entirely of triangular numbers. The same work also exhibits infinite families of a that admit such pairs and infinite families of a that admit none.
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The paper's central discovery is a rigid dyadic frequency law: for every k ≥ 0, the multiset {F(s) : 2^k ≤ s < 2^(k+1)} equals {3 + ν_2(j) : 1 ≤ j ≤ 2^k}, where F(s) counts occurrences of the odd value 2s-1 in the perturbed Hofstadter sequence. Equivalently, frequency r appears 2^(k-r+2) times for 3 ≤ r ≤ k+2, and the maximal frequency k+3 appears once. The paper also proves the dyadic endpoint id
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The central claim is Theorem 1: if s is a Somos sequence of order 6 with nonzero terms, then every 5×5 diamond minor of s×s vanishes, so the diamond rank is at most 4. The new order-7 analogue (Theorem 3) says every 5×5 half-diamond minor vanishes, where half-diamond means one of the two offset families is congruent modulo 4; these bounds also hold generically for twinned pairs (Theorems 2 and 4).
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The announced result is a non-definability theorem: for every odd q≥3 and d≥1 with q+d a power of 2, the reachability relation R(x,z) of the generalized Collatz map T_{q,d} — the relation 'z is some iterate of x' — has no first-order definition in the structure ⟨N,+,V_q⟩, where V_q(x) is the largest power of q dividing x. Because first-order definability in this structure coincides with recognitio
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The paper's central claim is that the arbitrary-step transition relation of the generalized Collatz map T_{q,d} is not first-order definable in base-2 Büchi arithmetic. The announced mechanism: definability of that reachability relation would put the exponential set P_q={q^y:y∈N} inside the same logic, where semilinearity forbids exponential growth. The supplied body instead shows that, for each f
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On the paper's own terms, the discovery is Theorem B: every map of the form f(n)=f0(n) for odd n, n/2 for even n — in particular the Collatz, Syracuse, and Baker maps — has only finitely many periodic orbits. The route is a 'dictionary': after defining the coarsest topology containing all two-point sets {n,2n}, the Borel structure makes f measurable, periodic orbits are open, every f-invariant pro
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On the paper's own terms, the discovery is Theorem B: every map of the form f(n)=f0(n) for odd n, n/2 for even n — in particular the Collatz, Syracuse, and Baker maps — has only finitely many periodic orbits. The route is a 'dictionary': after defining the coarsest topology containing all two-point sets {n,2n}, the Borel structure makes f measurable, periodic orbits are open, every f-invariant pro
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For derangement numbers D_n, the paper proves that D_{m+n}/n! equals a double sum over l and k of binomial coefficients, a sign, and a factor k!/l! times D_l. For harmonic numbers H_n, it proves the binomial-weighted identity binom(m+n,m) H_{m+n} = sum_{k=0}^n H_k binom(m+n-k-1,n-k) + H_m binom(m+n,n), and the degenerate version replaces the last binomial's top by m+n-λ and uses degenerate harmoni
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The central claim is that for n ≥ 3, a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)) takes only the value 1 or prime values, where b(n) obeys b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The paper further states that b(n−3) + n b(n−4) can be replaced by (n−1)! in the gcd without changing the result, yielding a finite continued fraction representation and a connection to the left fa
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On the paper's own terms, the discovery is that the Kurepa factorial is governed by the polynomial F_n(x) = sum_{k=0}^n k! x^k, obtained from the Fubini polynomial by fixing the Stirling number S(n,k) at k=1 (or k=n). Setting x=1 recovers the left factorial !(n+1), and its half r_n = F_n/2 is odd for n≥3. Theorem 16 then claims gcd(F_n,(n+1)!) = gcd(2r_n,2T) = 2·gcd(r_n,T) = 2, using the binary GC
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On the paper's own terms, the central discovery is Theorem 2.2: for every $m,n\ge 0$ and $\lambda\ne 0$, $$\mathrm{Bel}_{n+m,\$\lambda$}(t)=\sum_{k=0}^{m}\sum_{l=0}^{n}(1)_{k,\$\lambda$}\binom{m}{k}_{\$\lambda$}\binom{n}{l}$t^{{k}}$$F^{{(k)}}$_{n-l,\$\lambda$}(-\$\lambda$ t,k-m\$\lambda$)\,\mathrm{Bel}_{l,\$\lambda$}(t),$$ where $\mathrm{Bel}_{n,\lambda}(t)$ are the fully degenerate Bell polynomia
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The central claim is Theorem 1.1. For every n≥1 there are unique multivariate polynomials ψ_ν(x_1,...,x_ν) ∈ Z[x_1,...,x_ν] with no constant term, computable recursively, such that for every odd prime p>n, W_p ≡ Σ_{ν=1}^{n} p^{ν-1}/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^n, and equivalently (p-1)! ≡ -1 + Σ_{ν=1}^{n} p^ν/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^{n+1}. Here Q_p(ν) is the ν-th power sum of Fermat quo
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Starting from the generating function for the unsigned new type degenerate Stirling numbers of the first kind, the paper proves Theorem 2.1: these numbers equal Σ_{m=k}^n λ^{n−m} binom(n,m) [m k]. The main result, Theorem 2.2, states that for n≥k≥1, [n k]^*_{−λ} = Σ_{m=k}^n λ^{m−k} binom(n,m) S1(m,k) E[(S_k)_{n−m,λ}], where S_k is the sum of k independent products of a uniform(0,1) and an exponent
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The discovery is an operational substitution rule: if Σ t^n/n! a_n(x)=A(t)e^{xt}, then a_n(x)=A(∂_x)x^n. Writing the Padé approximant [r|s]A(t)=P_r(t)/Q_s(t) and replacing t by ∂_x gives [r|s]a_n(x)=P_r(∂_x)Q_s(∂_x)^{-1}x^n, an Appell-type approximation of a_n. Expanding the inverse denominator as a formal series turns this into combinations of truncated exponential polynomials; for Hermite amplit
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The central claim is Corollary 3.2: for every n ≥ 1, exactly 2^{n−1}(2^n−1) restricted colored base-3 partitions of (3^n−3)/2 satisfy condition (1.1). In particular, when n is prime and 2^n−1 is also prime, that count is a perfect number; conversely, by the Euclid–Euler characterization, every even perfect number is obtained by choosing n equal to a Mersenne-prime exponent. The paper explains the
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For a fixed ℓ, a multiple almost-Riordan array is the lower-triangular matrix whose column 0 has generating function b(t) and whose remaining columns are tg times cyclic products of ℓ multiplier series f1,...,fℓ. The paper asserts that these arrays close under the multiplication rule (37), with identity (1|1;t,...,t) and inverse (38), forming the multiple almost-Riordan group MaR. The central sequ
Stated claims
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The main discovery is that for every s≥0 and k≥1, the least eventual period T_k(s) of f_n(s) modulo 3^k is 1 when k ≤ ρ(s), and 2·$3^{{max(0, k-τ(s))}}$ when k > ρ(s), where ρ(s) and τ(s) are valuations defined from linear factors of s. The least starting index N_k(s) is given by an explicit table based on s mod 9; for the generic case it equals k + M(s) where M(s) is a small integer (0 or 1) depending only on s modulo 9, with a one-index reduction when a certain finite-field coefficient vanishes. These formulas are proven minimal by comparing 3-adic valuations of coefficients in a signed binomial expansion of the tail.
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The discovery is Proposition 15: for the alternating constant recurrence with parameters $a,b,c$, the coefficient array of the polynomials is the Sprugnoli array $(g,f_1,f_2)=\bigl((1-ax+cx^2)/(1+(a^2+b+c)x^2+bcx^4),\; x/(1-ax+cx^2),\; x/(1+(a^2+b+c)x^2+bcx^4)\bigr)$, with the internal relation $f_2=gf_1$. Its inverse, the moment array, is again a Sprugnoli array $(w,s_1,s_2)$ with $s_2=ws_1$, whose first column $w(x)$ is the moment generating function and whose production matrix is the two-periodic Jacobi matrix with diagonal alternating $a$ and $b-a$ and subdiagonal alternating $c$ and $b$. The paper then reads off the moment generating function as a Jacobi continued fraction, computes the Hankel transform $h_n=b^{\lfloor(n+1)^2/4\rfloor}c^{\lfloor n^2/4\rfloor}$, and applies the Stieltjes-Perron transform to obtain the weight function $\sqrt{-x^4+2x^2(a^2+b+c)-a^4-2a^2(b+c)-(b-c)^2}/(2(x-a))$ supported on the two intervals $[\pm\sqrt{a^2+b+c\mp 2\sqrt{bc}}]$. Proposition 22 extends the pattern to period three: with parameters $u,v,a,b$ and auxiliary constant $c=-(a+b+u^2+uv+v^2)$, the coefficient array is a four-series Sprugnoli array whose Jacobi matrix has the three-periodic striped form and whose support is determined by the zeros of an explicit degree-six polynomial, including families orthogonal on several arcs of a circle. The paper itself notes that the period-three case awaits further study, and it exhibits a family whose Jacobi matrix has the same striped shape but whose coefficient array is not a Sprugnoli array when the auxiliary condition fails.
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The paper's central claim is that the stabilizer problem for Chebyshev zero sets is completely soluble, with the answer dictated by the base-$p$ digits of $2d$ and $4d$. Theorem 1.1 states: for $k$ algebraically closed of odd characteristic $p$ and $d > 4$ with $p \nmid 2d$, the group $H_d = \{\tau \in \mathrm{PGL}_2(k) : \tau(\mathbb{Z}_d) = \mathbb{Z}_d\}$ equals $\{\mathrm{id}, \sigma\}$ with $\sigma(X) = -X$ precisely when neither $2d - 1$ nor $4d - 1$ is a power of $p$. Proposition 5.2 sharpens the first exception to $H_d \cong D_d$ when $2d = p^r + 1$, and Proposition 5.3 sharpens the second to $H_d \cong S_3$ when $4d = p^r + 1$; the theorem's proof shows these are the only routes to a nontrivial stabilizer. The engine is a residue dichotomy: any non-central element $\mu(X) = u + s/(X+u)$ of $H_d$ must satisfy the logarithmic-differential identity $\omega = 2(d-1)\,dX/(X+u) + \mu^*\omega$ with $\omega = -X\,dX/(X^2-4)$, and comparing the simple poles on both sides forces either $2d = 1$ or $4d = 1$ in $k$. Lucas' theorem converts those characteristic equations into the arithmetic conditions $2d = p^r + 1$ and $4d = p^r + 1$, and the classification of finite subgroups of $\mathrm{PGL}_2$ then identifies the exact groups.
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The central claim is that the dyadic divisibility of Tribonacci numbers at powers of two is exactly linear in the exponent: Theorem 3.0.4 states $v_2(T_{2^k}) = k+1$ for all $k \ge 5$. The proof reads $T_n$ off the companion matrix $A$ of the recurrence through $T_n = e_2 A^n e_3$, then decomposes $A^{2^k} = I + 2^{k-1}C_k$ with $C_k \equiv B \pmod 2$, obtained by iterated binomial squaring starting from $A^{16} = I + 8B$. Since $e_2 B e_3 = 2^3\cdot 7$, the target term becomes $T_{2^k} = 2^k(2^2\cdot 7 + \alpha_k)$, and the theorem reduces to showing $v_2(2^2\cdot 7 + \alpha_k) = 1$ for every $k \ge 5$. The supporting matrix work includes explicit formulas for $u^n$ and $u^{-n}$ in terms of $T_n$, $B_n$, and the signed sequence $R_n$ (defined by $R_n = -R_{n-1}-R_{n-2}+R_{n-3}$ with $R_0=R_1=0$, $R_2=1$), a Catalan-type identity $T_n^2 - T_{n-1}T_{n+1} = R_n$, Cassini-type relations obtained from $u^n u^{-n} = I$, and the trace identity $\operatorname{Tr}(u^n) = 3B_n + B_{n-1}$. The final section proves that every 3×3 matrix over $\mathbb{Q}$ with characteristic polynomial $x^3-x^2-x-1$ is similar to a companion matrix, hence all such Tribonacci-generating matrices are pairwise conjugate in $GL(3,\mathbb{Q})$.
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The central claim is Theorem 1.1: there are fixed real numbers $\alpha>1$ and $\beta=-\alpha/2$ such that, writing $\gamma=1/\alpha$, we have $\inf_{Q\in\mathbb N}Q^2\|Q\gamma\|>0$, and there is a sequence of positive integers $N_j\to\infty$ with $$\$int_0^{1}$\Bigl|\sum_{\substack{1\le n\le N_j\\n\in\mathcal B(\$\alpha$,\$\beta$)}}e(n\$\theta$)\Bigr|\,d\$\theta$\gg(\log N_j)^2.$$ Since Wooley's theorem gives a uniform $O_{\alpha,\beta}((\log 2N)^2)$ upper bound for such $\gamma$, this construction pins down the true growth rate at the $L^1$ endpoint and settles the proposed logarithmic bound in the negative.
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The central discovery is that the retriangular numbers (Theorem 2.1) are precisely the odd squares (2c+1)^2 for c >= 1, with b = T_c = c(c+1)/2, and this is the only possibility. This is extended to all s-gonal numbers (Theorem A): for s ≠ 4, the re-s-gonal numbers are the squares $x^{2}$ with x > 1 and x ≡ 1 (mod M_s), where M_s = 2(s-2)/gcd(s,4). Theorem B shows that there exists a non-constant affine map from s-gonal to t-gonal numbers if and only if M_t divides M_s. Theorem D reveals that the monoid of self-maps of P_s under composition is isomorphic to the multiplicative monoid H_{M_s} = {x ∈ N : x ≡ 1 (mod M_s)}, and the factorization into irreducible self-maps is unique only for triangular (s=3) and square (s=4) numbers.
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The central claim is Theorem 4.27 for even $m$, Theorem 5.5 for odd $m$, and Theorem 6.9 for centered $m$-gonal numbers: for every $m \ge 3$ and every prime power $q$, there is an explicit polynomial family in the triangular form $G_i = X^{d_i}Y_i - H_i(X, Y_0, \ldots, Y_{i-1})$ whose $\mathbb{F}_q$-point count equals $P_m(q)$, and a final construction whose count equals $C_m(q)$. The counting engine is Theorem 3.1, which splits every solution set into $q(q-1)$ points with $X \ne 0$ plus $q$ times the number $N_0(G)$ of common zeros of the associated family obtained by setting $X = 0$; the whole problem therefore reduces to prescribing $N_0(G)$. For even $m$, two operators, $L$ and $E$, raise $N_0(G)$ by exactly the powers of two required along the parent chain $m \mapsto m - 2^{\lfloor \log_2(m-4) \rfloor}$ that terminates in $\{4, 6, 8, 10\}$, whose four base families are built from the indicator-like polynomial $Y_0^{q-1}$. A corollary of the framework shows that only even $m$ can be realized by the bare triangular form for all $q$; odd $m$ are realized by adjoining the auxiliary equation $\Theta(X, Y_0) = 0$, which subtracts exactly $q(q-1)/2$ points from the even case $m+1$ and thereby produces $P_m(q)$. For centered numbers the same machinery realizes $C_m^*(q) = mq(q-1)/2$, and one extra equation adds the missing central point.
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The main result is Theorem 3.4: for every a,b ≥ 1 and 0 ≤ x ≤ y, the remoteness R(x,y) of position (x,y) in WYT(a,b) is 2 * min( iota_{a,b}(x), floor((y-x)/a) ) + epsilon, where epsilon is 0 at P-positions and 1 at N-positions. Here iota_{a,b}(x) is the smallest index m of a P-position (x_m, y_m) such that x_m or y_m lies in (x-b, x]. This reduces the selection of a fastest winning move to taking the smaller of two easily computed indices, whereas previous algorithms required searching among at most six candidate P-positions. When b=1, iota_{a,1}(x) equals the unique index of x in the Beatty sequences for α_a and β_a = α_a + a, giving an explicit formula entirely in terms of floor functions. The paper further shows (Theorem 5.2) that for any finite impartial game where each attempted move independently succeeds with probability p, the optimal winning probability is w_p(s) = (1 - (-λ)^{R(s)})/2 with λ = p/(2-p), and the optimal attempts are exactly the remoteness-decreasing moves, independent of p for 0<p<1.
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Theorem 1.4: For any non‑negative integer g, the least common denominator bD_g of all genus g Hodge integrals equals the Minkowski number T_{2g}, defined by its p‑adic valuation ν_p(T_{2g}) = Σ_{i≥0} ⌊2g/((p−1)p^i)⌋. The proof has two directions. The upper bound bD_g | T_{2g} is obtained by constructing finite Galois covers of M_{g,n} using abelian level structures of order N ≥ 3, whose degree is |Sp_{2g}(ℤ/Nℤ)|, and then taking the greatest common divisor of these degrees over all primes q ≥ 3. The lower bound T_{2g} | bD_g is established by expanding (t/2)/sin(t/2) in powers of t and using the Faber–Pandharipande formula to relate the coefficients to specific Hodge integrals; combinatorial lemmas show that the denominator of the coefficient of $x^{{q+r}}$ in the expansion has p‑adic valuation at least ν_p(T_{2g}), and that this coefficient actually achieves that bound.
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The paper's central claim is that discrete structure precedes algebra: two scalar primitives, $E_Z[m] = \exp(2\pi i m) \equiv 1$ and $E_{\bar 1}[m] = \exp(2\pi i) \equiv 1$, together with the operations of pointwise multiplication, inversion, and multi-branch rational roots taken in an unwrapped affine phase lift, generate the complex closure $G \cong (\mathbb{Q}/\mathbb{Z})^2$ and then, via spatial translation, sequence involution, and transverse Cayley–Dickson coupling, the quaternionic and octonionic sequence spaces. The proof mechanism is Lemma 3.3, which derives a transverse generator $J$ with $J^2 = -1$ and $I \star J = -J \star I = K$ from the interplay of translation and the involution $\sigma(\Omega)[m] = \Omega^*[-m]$, and Lemma 3.5, which derives a further generator $\ell$ with $\ell^2 = -1$ making the sequence associator $[\Omega_1, \Omega_2, \Omega_3] = (\Omega_1 \star \Omega_2) \star \Omega_3 - \Omega_1 \star (\Omega_2 \star \Omega_3)$ non-vanishing on non-Fano triads. The resulting closure $\mathcal{M}_{\mathbb{O}}$ is claimed to be an infinite divisible Moufang loop that is strictly associative on pairs, by Artin's theorem, yet carries genuine three-wave curvature, and that terminates at $\mathbb{O}^{\mathbb{Z}}$: sedenionic zero-divisor sequences would allow two nonzero wave packets to multiply to zero, destroying invertibility and norm multiplicativity. On this foundation the paper asserts three results—the $n=6$ invariant hexagonal mesh tied to the $G_2$ root system and its $\mathrm{SU}(3)$ stabilizer, the Catalan-tree phase jitter, and the rational loop closure dichotomy—and defines a real-valued sequence inner product proposed to make the seven Fano primitives orthonormal while bypassing the Jordan–von Neumann–Wigner obstruction.
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The central claim is Theorem 1.1: $\lim_{x\to\infty} \sup_{\varepsilon} (1/x)\,|\{1 \le n \le x : v_n(\varepsilon) \ne v_{n+1}(\varepsilon)\}| = 0$, where the supremum runs over all sequences $\varepsilon \in \{-1,1\}^{\mathbb{N}}$. Every sign sequence therefore has equal consecutive denominators on a set of asymptotic density one, and even the worst sequence over all sign choices has only $o(x)$ exceptions. The proof decomposes the exceptions by the prime $p$ whose contribution to the denominator changes, uses Lemma 2.1 to show such an index must be divisible by $p$, and then treats the primes in three ranges: fixed small primes by a dyadic interval iteration (Lemma 2.2), intermediate primes by a short-gap bound (Lemma 2.4) fed by polynomial root counting, and large primes by the fact that each candidate value is a fixed nonzero integer of size at most $x^L$, so few large primes can divide it.
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For any Sheffer sequence with exponential generating function $A(t)\exp(xB(t))$, fixing Padé approximants $R_A(t)=[m_A|n_A]A(t)$ and $R_B(t)=[m_B|n_B]B(t)$ defines a polynomial sequence whose generating function is $R_A(t)\exp(xR_B(t))$. The paper proves that this sequence is itself a Sheffer sequence, uniquely associated with the invertible functional $N^{[P]}$ and delta functional $L^{[P]}$ whose indicators are $1/R_A(R_B^{-1}(t))$ and $R_B^{-1}(t)$, respectively. It satisfies the biorthogonality relation $\langle N^{[P]}L^{[P]k} \mid s_n^{[P]}(x)\rangle = n!\delta_{n,k}$. The authors give explicit formulas for the connection coefficients $c_{n,k}$ linking the Padé–Sheffer basis to the original Sheffer basis, and show that the two sequences coincide for degrees $n \le M = \min(m_A+n_A,\, m_B+n_B)$. The rational form of $R_B(t)$ turns the Lagrange inversion formula into the extraction of coefficients from powers of a rational kernel, yielding a finite multinomial computation.
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The central claim is Theorem 2.1: for $q\ge1$, the $(0,q+2,1,2)$-hiccup sequence $a_q$ misses infinitely many positive integers, and if $c_q(1)<c_q(2)<\cdots$ lists them while $C_q(n)$ counts those at most $n$, then $a_q(n)=n+q+C_q(n)$ and $c_q(j+1)-c_q(j)=\lceil j/q\rceil$; the gaps are therefore $1,\dots,1,2,\dots,2,3,\dots$, each value repeated $q=x-2$ times. From this the paper derives the summation formula for $c_q(j)$, the closed counting formulas (2.6)-(2.7), the radical formulas $a_q(n)=\lceil(\sqrt{n}+\sqrt{q/2})^2\rceil$ valid exactly for $1\le q\le7$, the $q=8$ case repaired by adding 1 at the odd squares, an explicit error bound on $a_q(n)-n-\sqrt{2qn}$, the generating function $H_q(z)=\frac{z}{2}\bigl(\sum_{r=0}^{q-1} f(z^r,z^{q-r})-(q-1)\bigr)$ in terms of the bilateral $\theta$ function $f(a,b)=\sum_{\nu\in\mathbb{Z}} a^{\nu(\nu+1)/2}b^{\nu(\nu-1)/2}$, the nested recurrence $C_q(n)=q+C_q(n-C_q(n-1))$ for $n>q$, and the identification of $c_q$ with the partial complement $\hat P_q$ of the $(q+2)$-gonal numbers. The orbit of 1 under $a_q$ is the sequence of $(q+2)$-gonal numbers themselves. The flagship identity, Corollary 5.3, states that if $b_q$ enumerates the positive integers that are not $(q+2)$-gonal, then its $q$-fold iterate satisfies $b_q^{\circ q}(n)=a_q(n)-1$. The paper extends the nested recurrence and a postorder tree model to every miss increment $k+1\ge2$ and initial value $x\ge2$, and it reconnects the sequences to polyomino vertices, monochromatic edge counts, and core numbers of edge-coloured complete graphs.
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On the paper's own terms, the central claim is that the number of $m$-part sum systems, $N_m(n)$, has the Dirichlet series $\sum_{n\ge 1} N_m(n)/n^s = \sum_{L\ge 1} m!\,S(L-1,m-1)\,(\zeta(s)-1)^L$ with an explicit convergence abscissa $r_m$, the unique real number above 1 solving $\zeta(r_m)=m/(m-1)$, which grows like $\log m$. The argument rests on the identity $P(n,x-1)=Q(n,x)$ between the generating polynomials of the non-trivial divisor functions $c_j$ and the balanced divisor functions $c_j^{(-j)}$, so the two families are the same family shifted by one in $x$. Along the way the paper obtains rational-function Dirichlet series (3.3)--(3.6) for these polynomials, their sum-over-divisors recurrences, and, for squarefree $n$, their identification with ordered Bell polynomials and their signed variants, which places Bernoulli numbers as integrals of these polynomials and yields a formula for $\zeta(2t)$ in terms of balanced divisor values.
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The central claim is identity (2): for any smooth $f$ and any weights, $A_p[w_1,\dots,w_N] = c(p)\cdot\mathrm{Wronsk}(w_1,\dots,w_N)\cdot\partial_x^p$, where $A_p$ is the signed sum $\sum_{\sigma\in S_{2p}}(-1)^\sigma w_{\sigma(1)}\partial_x^p\circ\cdots\circ w_{\sigma(N)}\partial_x^p$ and the Wronskian is the determinant of the matrix of iterated derivatives of the weights. The paper computes the universal constants by specialising to the monomials $1,x,\dots,x^{2p-1}$: after that substitution a permutation contributes precisely when it is late-growing, and the constant is the signed sum $c(p)\prod_{k=0}^{2p-1}k! = \sum_{\sigma\in\Phi_p}(-1)^\sigma\prod_{k=1}^{2p-1}(E_k)_p$ of falling factorials $(E)_p = E!/(E-p)! = E(E-1)\cdots(E-p+1)$. Evaluating this formula gives $c(1)=1$, $c(2)=2$, $c(3)=90$, $c(4)=586656$, and so on up to $c(18)\approx 4.881\times10^{462}$, a set of values that matches no existing entry in the OEIS and is therefore registered there as the new sequence A392714. The growth is $\log c(p) \simeq 2p^2\log p - bp^2 + o(p^2)$ with subleading constant $b\geq 2.6744$, and the algorithm that reaches $p=18$ exploits the fact that the recursion's state is just the set of not-yet-placed labels, stored as bit masks, with all arithmetic done modulo primes.
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On the paper's own terms, the discovery is Theorem 1.2: let $w$ be a $C^3$-word over $\{1,2\}$ with $|w|>16$; then $w$ is a shuffle square if and only if $|w|_1 \equiv |w|_2 \equiv 0 \pmod 2$. The constant 16 is sharp, and the exceptional set is exhaustive and permanent: exactly 34 nonempty even-Parikh $C^3$-words are not shuffle squares, all of them smooth, so they are the exceptions in every class $C^k$ with $k \ge 3$ and in the smooth class. The same method yields Theorem 5.7 for $C^2$ (sharp threshold 48, exactly 212 exceptions, the longest sitting at lengths 46 and 48), Theorem 5.8 for $C^1$ (no global parity threshold, but a sharp reduction: every even-Parikh non-shuffle-square of length at least 36 has a proper nonempty shuffle-square prefix and suffix, exactly 230 words have no such prefix, and the bad-prefix tree is eventually the union of 422 periodic rays with periods 1122 and 2211), and Theorem 1.1 for the Kolakoski word (a prefix $P_m$ is a shuffle square if and only if $m$ is an even-Parikh length other than 4 and 8). The paper also characterizes morphisms that reflect shuffle squares in terms of odd powers of private symbols and run-length parity, and, for doubly binary (that is, $C^1$) words, gives the exact deletion distance to a shuffle square, the largest-twin size, and sharp windows for single repairs near an endpoint.
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Conjecture 3 of Kimberling is true. The paper proves that for every v ≥ 1 there are infinitely many integers N with a(N) = v, where a(N) counts the rows of the array floor(kφ^n) that contain N. The proof starts from a single witness, N_v = L_{4v-5}, shows via an exact description of the rows containing an odd-indexed Lucas number that a(N_v) = v, and then applies a Lucas shift lemma: adding a sufficiently large even-indexed Lucas number L_b to N does not change the set of rows containing it, provided N + 1 is not itself an even-indexed Lucas number. Since the witnesses satisfy this condition — no number of the form L_{4v-5} + 1 is a Lucas number — each v inherits an infinite family N_v + L_b with exactly v occurrences. A parallel result, derived from the Skolem–Bang theorem, gives the complete disjointness rule R_i ∩ R_j = ∅ exactly when j - i is odd and divides i, shows that merged consecutive rows rank into the Wythoff sequences, and extends the same rule to floor(kx^n) for any quadratic unit x of norm −1 including x = 1 + √2.
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The central claim is that the reduced-symmetry hierarchy defined by the recursion $x\partial_x^p F = (p-1)^{-2} h^{p-2}\partial_h^{p-1}(h^{3-p}\partial_h F)$ is algebraic at every order, with minimal polynomial degree $2(p-1)$ and generic Galois group $S_{2(p-1)}$, while the self-dual hierarchy defined by $x\partial_x^p L = h\partial_h^p L$ is transcendental for $p \ge 4$. The supporting evidence is exact: sextic and octic equations at $p=4$ and $p=5$ with generic Galois groups $S_6$ and $S_8$, and a degree-ten specialization at $p=6$ with group $S_{10}$. The two hierarchies are not independent: the last reduced companion $\Phi_q$ equals $D_p S_p$ for an explicit differential operator $D_p$ of order $p-3$, so each self-dual function is an algebraic primitive of an algebraic function, and algebraic primitives need not be algebraic. All companions share a principal symbol, producing a universal characteristic curve $A_p = 0$ that is rationally parametrized and carries a projective $S_3$ action; the discriminants factor as $A_p^3 B_p^2$ with monomial powers $x^{(p-1)(p-2)} H^{(p-2)^2}$. The paper further shows why $p=3$ is exceptional: the lower-order Euler polynomials coincide, restoring full scalar $S_3$, so the geometry lifts to a tetrahedral $S_4$ quartic field whose six branches are its edges.
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The central claim is Theorem 2.1: for any real $c, x$ with $cx(x+1)\neq 0$ and any sequence satisfying $(n+1)^r u_{n+1} = P(n)u_n - c n^r u_{n-1}$ with $r \geq 0$ and $P$ a polynomial of degree less than 4, the weighted sum $\sum_{n=0}^{p-1} P_r(n,x) u_n^2/(cx)^n$ differs from the boundary term $(p^{2r}u_p^2 + \frac{c}{x}p^{2r}u_{p-1}^2 - p^r s_p(x)u_pu_{p-1})/(cx)^{p-1}$ by a constant, with the auxiliary polynomial $s_n(x)$ given explicitly from the coefficients of $P$. This one identity subsumes the Legendre identity (1.5) and the Li–Sun identities (1.6)-(1.7), supplies new closed-form identities for Apéry numbers, Domb numbers, Almkvist-Zudilin numbers, Franel numbers and their relatives, and combines with ten known congruences to yield new supercongruences modulo $p^k$.
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The central discovery is the precise exponent of p that must divide any integer group determinant for ℤ_p^n when n is small. Theorem 2.1 gives the exponent for p ≥ 5 and n = 2,3,4,5; for instance, when n=2 the exponent is p+3, meaning $p^{{p+3}}$ | M_G(F) whenever p | M_G(F), and this is sharp. Theorem 2.2 does the same for p=3 for all n, with exponent 2·$3^{{n-1}}$. Moreover, the paper provides a full description of S($ℤ_3^{3}$) in Theorem 5.1, listing exactly which values divisible by $3^{{18}}$, $3^{{19}}$, and $3^{{20}}$ occur and under what conditions on the coprime factor.
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The paper proves that for every integer n>1, ν_2(Φ_n(2)−1) = n/rad(n) and ν_2(Φ_n(2)+1) equals lpf(n) when n is squarefree and 1 otherwise. These extractors arise from the Möbius product representation of cyclotomic polynomials and radical reduction. Furthermore, by studying the logarithmic discrepancy log_2 Φ_n(2) − φ(n) and controlling the tail of the correction series, the paper develops a real reconstruction procedure that recovers each prime divisor of n by successive rounding. For odd squarefree indices the rounding is to the nearest odd integer. The same tail bounds permit passage to the limit over odd primorials, defining a constant Ω = 0.25061403238015047218… such that the recursion q_{j+1} = OddRound(S(Ω−1+F_1({q_1,…,q_j}))) yields every odd prime, starting from the empty set. The paper also derives a Mellin transform identity linking the underlying series to the Riemann zeta function.
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The central discovery is that the finite harmonic q-series E_{(p-1)/2}(q_p) equals, modulo (1 - q_p), the quotient of the two Apéry solutions evaluated at index p-1, scaled appropriately. This yields the explicit congruences A_{p-1} ≡ -3p (mod $p^{2}$) and C_{p-1} ≡ -2p/21 B_{p-3} (mod $p^{2}$), and consequently the algebraic limit (E_p mod (1 - q_p))_p = 3/4 Z(3) and the analytic limit lim_{N odd} E_N = 3/4 ζ(3). In particular, Tasaka's conjecture (C_{p-1}/A_{p-1})_p = 2/21 Z(3) follows as a corollary.
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The central discovery is Theorem 3 (comprising Theorems 19 and 23): for any d ≥ 1, any f ≥ 1, and any g ≥ max(f+1, d-1), the 3-sumfree sequence S_{f,g,g+d} is ultimately periodic. The period and preperiod sizes are expressed in terms of f and g, and the modulus of periodicity is a linear combination of f, g, and d. This result extends earlier work on 2-sumfree sequences and provides a unified framework for understanding periodicity in a large family of 3-sumfree sequences.
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Theorem 3.1 of the paper states that for any function f from primes to the natural numbers, if f(p)+pZ represents the Montgomery reduction REDC_{p,R(p)} (i.e., f(p) is the modular inverse of R(p) modulo p) for all but finitely many primes p, then the image of f in the ring A = (∏_p F_p)/(⊕_p F_p) is naively transcendental over Q. The proof reduces the problem to the case where f is the inverse of R(p), and shows that R(p) — a power of a fixed base B bracketing p — defines a transcendental element in A by Proposition 2.2, because the set S = {B^e} is sparse enough to satisfy the condition x log e_S(x) log e_S(x−k) ∈ o(e_S(x−k)). The transcendence of the inverse follows because the product of R(p) and its inverse is 1 in each component, and if the inverse were algebraic then R(p) would also be algebraic, a contradiction.
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Theorem 1.1: For every fixed integer K ≥ 2, the rank-capped Enots-Wolley sequence EW[2..K]—defined as the lexicographically earliest infinite sequence a_1=1, a_2=2, and for n≥3, a_n is the least unused integer in {m≥1 : 2 ≤ ω(m) ≤ K} satisfying the three local conditions (retain a prime from a_{n-1}, avoid primes of a_{n-2}, and introduce a new prime not in a_{n-1})—is a permutation of the set {1,2} ∪ {m ≥ 1 : 2 ≤ ω(m) ≤ K}. In other words, every integer with at least two and at most K distinct prime factors occurs exactly once.
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The paper establishes Theorem 1.1: there is a positive constant c such that the maximum element M_n and the largest prime Q_n completely present in the n-th closure both satisfy M_n ∼ Q_n ∼ c φ^n, and the cardinality |C_n| ∼ c $φ^{{n-1}}$. It follows that lim_{n→∞} |C_{n+1}|/|C_n| = φ and lim_{n→∞} |C_n|/M_n = 1/φ. The proof uses a Goldbach-type filling lemma to extend complete prime prefixes, a contracted recurrence argument to extract asymptotic constants, and a comparison of the two constants to force equality, ultimately deriving the golden-ratio growth.
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The central claim is Theorem 24: log N(n) = log A_adm(n) + o(n/log n) as n → ∞. Here N(n) is the number of distinct survivor sets obtainable from the triangular residue-class model (forbidding one class modulo every integer 2,…,n), and A_adm(n) is the number of subsets of {2,…,n+1} that omit at least one residue class modulo every prime. The theorem shows that two superficially different combinatorial counting problems have exponentially equivalent growth, with a relative error that is subleading on the scale n/log n. As corollaries, the paper establishes that log N(n) is of order n/log n with any leading constant between log 2 and 2 log 2, and that the prime-trace logarithm is asymptotic to (log 2) n/log n.
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The paper discovers that $9 \in U_4 \setminus D_4$, i.e., the 4-step gap set for up-integers contains 9 while the 4-step gap set for down-integers does not. This provides a concrete counterexample to the conjecture of Chung, Graham, and Spiro that $D_\ell = U_\ell$ for all $\ell \ge 1$.
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The central claim is Theorem 4: for each integer k≥6, there exists ε_k>0 and a k-AP covering set A such that |A∩[0,n]| < $n^{{(k-2)/(k-1)-ε}}$ for all sufficiently large n. This directly refutes Conjecture 3 of [KSY18], which asserted a lower bound of the form c_k $n^{{(k-2)/(k-1)}}$ infinitely often. Theorem 5 gives a stronger exponent 1 - c/log log k for all sufficiently large k. Theorem 7 establishes that if the Arithmetic Kakeya Conjecture holds, then for any ε>0 and all large k, every k-AP covering set must satisfy |A∩[0,n]| > $n^{{1-ε}}$ for large n.
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The central discovery is that Fraenkel's conjecture is true: for any m ≥ 3, if the sets S(\alpha_i, \beta_i) partition Z and the \alpha_i>1 are pairwise distinct, then the set of moduli is exactly \{ (2^m-1)/2^j : 0 \le j < m \}. Equivalently, the densities are the binary fractions 1,2,4,\ldots,$2^{{m-1}}$ normalized by 2^m-1. The proof reveals a fundamental structural property: in any such partition, the minimum-gcd Fourier layer gives rise to a finite inverse-sine row system that cannot be all-passing when all densities are below 1/3. This yields the one-third theorem, and a subsequent combinatorial induction recovers the full pattern.
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Wirsching's Conjecture 1 (Theorem 4.3) is true: the pointwise lower bound on generators implies the same bound on Elka functions, with the same constant and a smaller window radius. Conjecture 3 (Theorem 10.1) is true: for any δ_5>0, φ(z_ℓ)/φ_0(z_ℓ) → $e^{{H(0)}}$ uniformly for sequences (z_ℓ) in the class $Ã^{{δ_5}}$, with H(0) certified in (0.53412203666478,0.53412203666479). Theorem 10.2 shows the unrestricted asymptotic φ(t) ∼ κ φ_0(t) fails because H is not constant. Condition (⋆4) holds with μ=1/3 (Theorem 11.1). Consequently, the chain from (⋆3) to positive predecessor density is now fully proved.
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For $\phi(x)=x^d$, the $n$-th dynatomic polynomial factorizes into cyclotomic polynomials: $\Phi_{d,n}(x)=\prod_{\operatorname{ord}_k(d)=n}\Phi_k(x)$, where $\operatorname{ord}_k(d)$ is the multiplicative order of $d$ modulo $k$ (Lemma 3.1). Pushing this factorization through the classical theorem that $\operatorname{Res}(\Phi_a,\Phi_b)=p^{\varphi(a)}$ exactly when $b/a$ is a power of a prime $p$, and $1$ otherwise (Theorem 2.2, stated without proof), the paper obtains the full resultant table. For $m>1$, $\operatorname{Res}(\Phi_{d,1},\Phi_{d,m}) = \Phi_m(d)^{d-1}$; for $1<n<m$, $\operatorname{Res}(\Phi_{d,n},\Phi_{d,m})=1$ if and only if $n\nmid m$; and when $n\mid m$, every prime $p$ dividing $\Phi_m(d)$ occurs in the resultant with valuation at least $v_p(\Phi_m(d))\cdot\sum_{k\mid n}\mu(n/k)d^k$. Because all these resultants are non-zero, points of formal period $n$ and $m$ can never coincide for $n\ne m$, and Corollary 4.5 states that for $x\mapsto x^d$, formal period $n$ implies exact period $n$.
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Starting from Nicomachus' original odd-number pattern, the paper's central discovery is that the cubic case is exceptional in two complementary ways. Locally, requiring $n^p=b^2-a^2$ with $b-a=n$ forces $a=(n^{p-1}-n)/2$ and $b=(n^{p-1}+n)/2$; these are exactly the consecutive triangular numbers $T_{n-1}$ and $T_n$ when $p=3$, since $b=T_n$ reduces to $n^{p-1}=n^2$. No other $p$ satisfies this for all $n$. Globally, matching degree and leading coefficient in $S_s(N)=S_1(N)^r$ yields $s+1=2r$ and $s+1=2^r$, so $r=1$ or $r=2$, giving only the trivial pair and Nicomachus' pair. Thus the identity $S_3(N)=S_1(N)^2$ is the unique nontrivial relation of this exact form.
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The central discovery is a theorem: for p a positive integer, any sequence satisfying a_{n+1}u_{n+1}(x)=b_n(x+d_n)u_n(x)-c a_n u_{n-1}(x) with a_0=0 and c\neq 0 obeys \sum_{n=0}^{p-1} \frac{b_n}{c^n}u_n(x)u_n(y)=\frac{a_p}{$c^{{p-1}}$(x-y)}\big(u_p(x)u_{p-1}(y)-u_{p-1}(x)u_p(y)\big), with the x=y limit giving a derivative identity. The paper also derives a companion identity with alternating signs. Specializing to the recurrences defining g_n and v_n, and using the identities G_n(x)=g_n($2x^{2}$+2x+1) and V_n(x)=v_n($2x^{2}$+2x+1), it obtains closed expressions for sums like \sum_{n=0}^{p-1}(2n+1)v_n(x)^2 and \sum_{n=0}^{p-1} g_n(x)^2, which it then expands p-adically. The expansions yield congruences modulo $p^{5}$ for weighted sums of v_n(x), modulo $p^{7}$ for sums of V_n/16^n, modulo $p^{4}$ for \sum (2n+1)V_n(x)^2, and modulo $p^{3}$ for \sum G_n(x)^2, confirming Z.W. Sun's conjecture (5.5).
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On the paper's own terms, the discovery is that degenerate special numbers are governed by a pair of operator identities. The first expands $\left(x^{1-\lambda}\frac{d}{dx}\right)^n$ as $\sum_{l=0}^n \binom{n}{l}_{\lambda}\, x^{l-n\lambda}\left(\frac{d}{dx}\right)^l$, and the second rewrites $x^{n\lambda}\left(x^{1-\lambda}\frac{d}{dx}\right)^n$ as the $\lambda$-deformed Euler operator $\left(x\frac{d}{dx}\right)_{n,\lambda} = \sum_{l=0}^n \binom{n}{l}_{\lambda} x^l\left(\frac{d}{dx}\right)^l$. Applying these operators after the substitution $u=e_{\lambda}(z)$, the paper derives Theorem 2.5, $E_{n,\lambda} = \sum_{k=0}^n \binom{n}{k}_{\lambda} (-1)^k k!\,2^{-k}$; Theorem 2.7, $2^n \beta_{n,\lambda/2} - \beta_{n,\lambda} = -\frac{n}{2} E_{n-1,\lambda}$; and Theorem 2.10, $\mathcal{E}_{n,\lambda} = \sum_{k=0}^n 2^{-k}\sum_{j=0}^k (-1)^j \binom{k}{j} (2j+1)_{n,\lambda}$. In each case the operator converts a derivative evaluation at zero into a finite weighted sum whose coefficients are the degenerate Stirling numbers of the second kind and whose weights are powers of $1/2$.
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The paper's central claim is Theorem 1: there is a constant $c>0$ such that $|A\cap[1,x]|\geq cx$ for all sufficiently large $x$. The proof does not try to list $A$ directly. It encodes compositions of the maps $T_a(x)=ax-1$ as paths through nineteen half-open intervals, switches among four assignments of the multipliers $2,3,5,9,14$ to keep a three-dimensional prime-exponent imbalance near the origin, and forces every return to a fixed state to have slope $q^m$. The arithmetic renewal theorem converts the distribution of return exponents into an asymptotic count of distinct paths, and disjoint inverse-image intervals give the injectivity that the non-free affine semigroup cannot provide globally. Evaluating the resulting maps at $17$ produces distinct elements of $A$ and yields the linear lower bound.
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The paper proves a dichotomy for square exponents: when $k=2$, the generalized equation (1.2) has integer solutions only when the left side is a sum of two or of four consecutive arithmetic-progression terms. In both cases there are infinitely many solutions, and every solution is 'good,' meaning the quotient $\lfloor (m-b)/q \rfloor$ does not change when $m-b$ is replaced by $m-b-1$; geometrically, the solution set consists of evenly spaced integer points along lines in $\mathbb{R}^3$, and scaling any solution by a positive integer produces another. The supporting argument shows that no three-term solution exists (via a modulo-3 infinite descent), that no solution with five or more terms exists (via a quadratic inequality), and that in the two- and four-term cases $x$ cannot divide $z-y$, which is what forces the goodness property. A separate theorem gives the asymptotic linear growth of $k$ in a fixed progression, and for $k=3$ the paper shows that three-term solutions are parameterized by rational points on the elliptic curve $W^2=R^3-648$.
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The central discovery is Proposition 13: when f(x) is an even power series, the square of the Sprugnoli array M(g,f) equals ((g(x)g(-x)+$x^{2}$ f(x)^2))/(1-$x^{2}$)^2, which is an aerated element of the Appell subgroup of the Riordan group. Consequently, every aerated Appell array that can be written in this form has a square root that is a Sprugnoli array M(g,f). If the right-hand side is the constant 1, then M(g,f) is an involution in the Sprugnoli group. The proof uses the Sprugnoli group product rule and the fundamental theorem of Sprugnoli arrays to compute the square componentwise.
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The central claim is Theorem 1.4: for $X$ and $Y$ independent of $q$, the unitizing operator of degree $3$ applied to $\frac{f_6}{f_2}(qX,q/X,qY,q/Y;q^2)_\infty$ returns its cube. The two auxiliary variables are not decorative; choosing $X$ and $Y$ among the fifth roots of unity $\rho,\rho^2,\rho^3,\rho^4$ produces the last pair of 3-convolutive products in Table 2, while the choices $1,-1,\omega,i,\zeta$ produce the other twelve primitive cases and several nonprimitive ones. The paper's second result, Theorem 1.2, is the analogous one-parameter identity for the 2-convolutive case: the degree-2 unitizing operator applied to $\frac{f_2 f_6}{f_1 f_3}(qX,q^2/X,q^2X,q/X;q^3)_\infty$ gives its square, and this specializes to the two newly discovered eta-products (II.5) and (II.6). Together with the earlier Theorem 1.1, this leaves only (II.7), an eta-product not reachable from either parametrization, as sporadic.
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The central claim is that the rational function (1+cxt)/(1+axt+bt^2) generates a genuinely unified family: K_n(x|-2,1,-1)=T_n(x), K_n(x|-2,1,0)=U_n(x), K_n(x|-2,1,-1/x)=V_n(x), K_n(x|-2,1,1/x)=W_n(x). The paper establishes the recurrence K_n = -ax K_{n-1} - b K_{n-2} with K_0=1 and K_1=(c-a)x, an n×n tridiagonal determinant formula, and a closed form as a sum of binomial coefficients. It further proves a closed form for T_n(x|a,b), identifies T_{2n}(x|1,-1) with the Morgan-Voyce polynomial b_n(x^2), identifies T_{n-1}(x|-1,-1) with the Fibonacci polynomial f_n(x), and expresses Fubini polynomials as F_n(x)=Σ_{m=0}^n K_m(x|-1,0,0) {n choose m} m!.
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The main result, Theorem 3.25, states that for all n≥k≥1, kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k). This answers affirmatively the 1985 question of whether the second-kind Comtet numbers admit a recurrence involving only a fixed number of terms. The proof derives from two elementary identities satisfied by the weighted Stirling polynomial G_{n,k}(x): the polynomial recurrence G_{n+1,k}(x)=(x+k)G_{n,k}(x)+kG_{n,k-1}(x) and the difference relation G_{n+1,k}(x+1)=G_{n+1,k}(x)+G_{n+1,k+1}(x). Evaluating these at x=-n and substituting the special-value identity G_{n,k}(-n)=k!B(n+1,k+1) yields the four-term recurrence after a shift of indices. The paper further proves the unsigned counterpart, gi
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The central claim is Corollary 5.3: for every integer k ≥ 0, Φ(z,s,a) = (8/3)H_k - 2H_{k+1} + (1/3)H_{k+2}, DΦ(z,s,a) = (-4H_k + 5H_{k+1} - H_{k+2})/(3·2^k), and D^2Φ(z,s,a) = (2H_k - 3H_{k+1} + H_{k+2})/(9·4^k). These follow from the operator identity H_k = (α_k D^2 + β_k D + 1)Φ, with α_k = (3/2)4^k and β_k = (3/2)2^k, combined with a Vandermonde inversion on the geometric factors 1, 2, 4. The paper also proves the convergence of the defining series, a reduction of H_k to shifted Hurwitz–Lerch functions, the recurrence H_{k+3} = 7H_{k+2} - 14H_{k+1} + 8H_k, a generating function in k, and special values connecting to Bernoulli, Eulerian, and Euler polynomials.
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For every even positive integer h and every B_h-set A, the liminf as n tends to infinity of A(n) divided by the h-th root of n/log n is at most (pi/log 2 times Gamma(1+h/2)^2 over Gamma(1+1/h)^h) raised to the power 1/h.
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The paper establishes that for every N≥1 and 0≤r≤N, the maximum of C_min(d) over binary words of length N with exactly r ones is attained exactly, up to rotation, by the Christoffel word d^chr_{N,r}. The functional C(d) is defined from the parity word as C(d)=Σ 2^{i−1}3^{r_i(d)} d_i, where r_i(d) counts ones strictly to the right of position i; this is the numerator in the standard affine formula expressing the N-th iterate as a function of x. For a periodic orbit, x = C(d)/(2^N − 3^r), so maximizing C_min over the class gives the largest possible numerator and therefore the largest possible minimum element of a cycle. The paper then derives two consequences: no periodic orbit can have N>2r,
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Framing every integer as a Mersenne word — digits 0,1,2 over weights 2^k − 1, with a 2 forcing all lower digits to 0 — yields two results at once. A successor transformation on the nonbinary sublanguage enumerates A055938 and gives the difference rule y_{n+1} − y_n = 3 − 2χ(n); this proves the conjectural identity A080578(n) = A055938(n−1) + 2 for n ≥ 2, and shows the binary-side counting function Z equals a shifted Conolly sequence. The same parent map π = r − Z, conjugate to digit deletion, builds an integer profile σ; the window-counting theorem says D(ξ) ∈ {0,1} and D(ξ)=1 ⟺ σ(ξ+K) − σ(ξ−Ω) ≡ 2 (mod 3). Since the finite-field polynomial M(a,b) detects 2 in its second argument, σ mod 3 sa
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On its own terms, the paper proves Theorem 1: if an H-invariant Legendre pair of length 333 exists, then H is a subgroup of the kernel of the reduction (Z/333Z)^× → (Z/3Z)^× and |H| ≤ 6. Equivalently, inside the order-108 kernel there are exactly 30 subgroups; 21 are proved impossible, including every subgroup of order at least 9. The decisive analytic step treats the final order-9 subgroup H12 = ⟨10,46⟩: because its mod-37 image has order 9, each column sum of the 9-compression lies in {±1, ±17, ±19, ±35, ±37}, the total squared norm 594 forces exactly two entries of magnitude 17, and then a single-shift autocorrelation bound contradicts the required compressed correlation −74. The paper al
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The central claim is the set of explicit formulas and Bell-identities in Section 3: e.g., M_n(x)=n!∑_{k=0}^n binom(n+k+1,n-k)x^k (Theorem 3.1/Corollary 3.3), and Theorems 3.4, 3.6, 3.7 asserting that binomial-type sums of Bell polynomials equal finite sums over K_n, J_n, L_n. If correct, these are new identities; if the printed Theorem 3.7 is taken as stated, it is false.
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On its own terms, the central claim is Theorem 4: under the hypotheses p_i(0)≠0, p2 coprime to p1 and p3, and d2 ≤ d1+1, the k-th column (k≥1) of the Riordan array (p1/p2, t p3) over F_q is ultimately periodic with least period π, the least period of the coefficient sequence of 1/p2; its periodic block is C_k = A^k C_0, with A a π×π circulant matrix generated by the coefficients of p3. Theorem 8 extends the picture to the 3-D array (p1/p2, t p3, p4): the orbit of periodic blocks in the k-th layer is M_4^k applied to the orbit in layer 0, M_4 a circulant built from p4. Theorem 10 adds that the sequence of preperiodic column partial sums is eventually periodic, and Corollary 11 exhibits a fami
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For every irrational α greater than (3+√5)/2 and every positive integer n there exists a necessary and sufficient elementary-number-theoretic condition that decides whether a natural number x belongs to the nth iterate of the Beatty sequence of modulus α.
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The central claim is Theorem 3.1: for c=(p+r-1)/p, the coefficient sequence B_{p,r,k} vanishes for k>p+r-c and satisfies the reflection B_{p,r,k}=B_{p,r,p+r+1-c-k} on its support if and only if r≡1 (mod p). The proof derives this by combining a generating-function identity with a root analysis of the polynomial g(X)=binom(pX,p+r), whose roots are exactly m/p for 0≤m≤p+r-1. Reflecting the variable X to c-X preserves these roots only when c is integral, which is equivalent to the congruence condition. The paper also proves that under the same congruence every coefficient is a multiple of p, and that B_{p,1,1}=p C_p, the p-th Catalan number.
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For every integer k≥1 and every n≥0 the Laurent coefficients J_{k,n}(m) of the normalized Jacobi tail are positive for all integers m with -(k+n)≤m≤k+n. Equivalently, the generating function of the tail equals z^{-k}(1+z+⋯+z^{2k})/((1-qz)(1-z^{-1}q))+z^{-k}N_k(z,q), where N_k has exclusively non-negative coefficients.
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The interlace and deinterlace operators of the proposed stream algebra are instances of Fraenkel partitions of the natural numbers (and therefore of Beatty sequences). Consequently the algebra is deterministic: every continuous query over constant-rate medical streams yields an exact, reproducible result that can be optimised by algebraic rewrites.
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The generalized q-Morgan-Voyce polynomials defined by M_n(x,q)=(x+1+q)M_{n-1}(x,q)-q^{n-2}M_{n-2}(x,q) admit generating functions and explicit formulas obtained via the Fibonacci operator and the binomial theorem, possess natural negative-index extensions, satisfy summation and determinantal identities, and obey a matrix-based q-Cassini identity.
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Every infinite exponent code generated by the accelerated Collatz orbit of a fixed positive odd integer forces both the 2-adic start residue rate and the 3-adic endpoint residue rate to vanish asymptotically. Near-critical codes produced by random sampling, mechanical construction, and adaptive evolutionary search at lengths 100–400 retain clearly positive rates on both residues, so they remain incompatible with any fixed finite starting value.
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For every integer k≥3 the sequence of kth powers is not maximal as a GD sequence; the sequences of integers congruent to 1 modulo k are maximal precisely when k=2 or 3 and are non-maximal for all k≥4; and the sequences of the form k^n-1 are non-maximal for every k≥3.
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There is a uniform recursion (Theorem 4.2) that expresses every Hankel determinant H_w(n) of a weighted binary sum-of-digits sequence in terms of two smaller determinants of the same type (or of the first-difference type). For ordinary sum-of-digits the recursion collapses on the indices n_k=⌈2^{k+2}/3⌉ to the elementary closed form (-1)^{(k+2)(k+3)/2}(k+1), giving an infinite explicit family.
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U(n) is odd if and only if n belongs to the even-offset class J, and even if and only if n belongs to the odd-offset class K; the successive elements of these classes are given by the closed forms j(n)=2n-⌊(√(8n+1)-1)/2⌋ and k(n)=2n+1+⌈(√(8n+9)-1)/2⌉. Consequently the parity of every Redheffer matrix is completely determined by evaluating one of these two formulas.
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For a broad class of classical sequences the dilated Hankel determinant ¨H_n(a)=det(a_{2i+j}) admits a simple product evaluation, even though no universal continued-fraction formula exists for it and the class of sequences with known product formulas is larger for the ordinary Hankel determinant. The evaluations are obtained by six methods developed in the paper and include, among others, the factorial, Catalan, Euler/secant, Springer, reciprocal-sine and Bessel families; they also settle a conjecture on the roots of the Poupard and Kreweras polynomials.
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In any modular g-Golomb ruler the cyclic gaps themselves obey the multiplicity bound of g. Consequently the largest gap is at least the ceiling of (N + ga(a−1)/2 + ra)/n, where n = ag + r. Cutting at that gap yields an ordinary g-Golomb ruler whose diameter is strictly smaller than the diameter obtained from the classical average-gap cut. The same extraction applies after a controlled folding that converts a modular Golomb ruler into a modular g-Golomb ruler.
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The paper introduces S_l^(m)(n) = sum over k_1+...+k_l=n of the m-th power of the multinomial coefficient, and records the identities S_2^(m)(n)=f_n^(m) (Franel numbers) and S_4^(2)(n)=D(n) (Domb numbers). It then proves Theorem 1.1: for p prime with p∤(l+1), the alternating sum ∑_{n=1}^{p−1} (−1)^{mn}/n^{m−1} S_l^(m)(n) ≡ 0 mod p, together with a q-analogue over cyclotomic polynomials. The centrepiece, Theorem 1.3, establishes the previously conjectured mod p^2 evaluation ∑_{n=1}^{p−1} D(n)/n ≡ (p/3)(2/5)p B_{p−2}(1/3) mod p^2. The proof reduces the Domb sum to coefficients of F(x)^5 and F(x)^3 built from squared factorials, and the final Bernoulli-polynomial term comes from a known congrue
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The central claim is that the natural density ε of Mersenne numbers satisfies ε × ω = 1, where ω = 1 + 1/3 + 1/5 + 1/7 + ... is the sum of reciprocals of odd natural numbers. This is derived by assigning density ε/(2x+1) to column x of a pairing-function matrix and requiring that all column densities sum to 1. The result is presented within the framework of nonstandard analysis, where ε and 1/ω are equivalent infinitesimals, and the conclusion is that the Mersenne density is strictly positive.
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When the six GKP parameters are treated as indeterminates, each row sequence (T(n,k))_k is coefficientwise strongly log-concave, and the sequence of row-generating polynomials (P_n(x))_n is coefficientwise strongly log-convex (hence Hankel-totally positive of order 2) jointly in x and the six parameters.
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Lemma 2.1 asserts that for a nonnegative integer n and complex z outside {0,…,n}, ∑_{k=0}^n (z choose k)(-1)^k (x+1)^k equals (-1)^n (n+1)(z choose n+1) times ∑_{k=0}^n (n choose k) x^k/(z-k). Every later identity in the paper is obtained from this equality by substituting particular values, replacing z by half-integers, integrating with respect to x, or differentiating with respect to z.
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There exists an explicit constant γ0 = 0.943492590... < 0.9435 such that the largest Sidon subset of {0,...,N-1} satisfies F(N) ≤ N^{1/2} + γ0 N^{1/4} + O(1). The constant is realized by a concrete eight-kernel rational certificate that produces a product ab strictly less than (0.9435)^2 inside the vector smoothing lemma.
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For consecutive fixed points of J4 at least 3, the pure high extremal points between them form a finite alternating sequence of type-1 and type-2 blocks whose lengths, last points, and exit to the next fixed point are completely determined by the ternarity of the starting fixed point, the valuations u3(4np+2) or u3(4np+3), the transition meters u3(4ne+3) and u3(4ne+5), and an explicit termination rule—without computing any intermediate values of J4.
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For any γ ≥ 1, no infinite set of integers with at most γ representations of every positive difference can be denser, in liminf, than 2√γ/√(log 2) ≈ 2.40√γ times √(n/log n). The proof is an energy argument: split the set into blocks of length N; the γ-Golomb condition bounds the sum of squared block counts from above by γN + o(N), while a weighted Cauchy inequality bounds the same sum from below by τ_N² (log 2)/4 · N + o(N), where τ_N is the minimal value of A(n)√(ψ(n)/n) for n ≥ N. The two bounds force τ_N ≤ 2√γ/√(log 2). The same paper shows every γ-Golomb ruler has limsup of A(n)/√n at most √γ, and constructs a γ-Golomb ruler achieving at least √γ/√2, so the limsup density is determined u
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In the odd-compressed Syracuse version of the Collatz map, upward phases occur according to a homogeneous Poisson process whose intensity is set by the average logarithmic growth rate and the geometric law of 2-adic valuations; the resulting count N↑ therefore follows a Gamma distribution whose scale θ equals 2/(2−log₂3)² and whose shape K grows logarithmically with the maximal initial value X₀ = 2L+1. Closure conditions on periodic orbits further limit nontrivial cycles, supporting the statistical description.
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The central claim is that for each m≥1, exactly F(m+1) odd integers in {1,…,2^m} avoid the residue class 4 (mod 6) on their orbit during steps 2 through m (Theorem 4.7). This is equivalent to saying that the modular encoding of the orbit never visits vertex 4 of the graph G' for those steps. The proof uses a spectral gap argument on the modular Collatz graph G' = G[{1,2,4,5}]: removing vertex 4 leaves the subgraph H4 = G'[{1,2,5}] whose adjacency matrix has characteristic polynomial λ($λ^{2}$-λ-1) and spectral radius φ, whereas ρ(G') = 2. The number of paths of length ℓ in H4 starting from {2,5} is exactly F(ℓ+3), and because the suffix (h_2,…,h_m) of the modular encoding is invariant under shifts for odd n (Corollary 4.4), these paths correspond bijectively to the odd integers in {1,…,2^m} that avoid vertex 4. The paper also shows that every positive cycle under T must visit vertex 2 (mod 6) and that at least about 18.45% of the steps in any positive cycle belong to that residue class (Corollary 4.19).
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We prove a sharp finitary form of Terras's parity-vector density. We give a closed-form analytic count of paradoxical Ω_k(n) for each fixed length k. We establish a density-zero theorem for bounded-length paradoxical sequences with explicit constant. Numerically, every paradoxical reduced ratio q/j is a left convergent, left semiconvergent, or Stern-Brocot mediant of adjacent convergents of log_3 2.
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In the base octave model every integer is written uniquely as n = B + 8(A-1) with B in 1 to 8. The Collatz rules induce parity-dependent transitions on B together with affine updates on A. Refining the parity description produces a finite 128-state symbolic system that encodes all admissible transitions including carry effects. Enumeration of return paths between persistence episodes shows that the sole unbounded persistence mechanism, occurring via base-7 transitions in even octaves, is necessarily bounded by the 2-adic valuation of the octave index. This forces a non-positive drift in the logarithmic octave coordinate and confines trajectories to the contractive subnetwork associated with
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The central result is a determinant identity: for each wug-snake graph, the determinant of its associated continuant matrix equals the weighted perfect matching number of that graph. This identity bridges the combinatorial world of graph matchings with the algebraic world of matrix semigroup invariants, specifically Markov-Davenport forms. By constructing polymino wug-tiles for matrices whose determinants compute these forms, the paper establishes that Markov numbers of matrices and matrix semigroups are encoded in perfect matching structures, generalising the classical snake graph correspondence that connects Markov numbers to domino tilings.
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The central claim is that the implicit equation a(Σ_{j=0}^{r−1}a(n−j))=n, for a nondecreasing positive-integer sequence a, has, besides the greedy solution, an explicit inhomogeneous Beatty solution a(n)=⌊n/√r+√r/2⌋ for every r≥2 that is not an even perfect square, and that this formula solves the equation for all n≥r. When r=(2m)², the same formula gives n+1 instead of n, and this failure is shown to be systematic. The proof route is: solve the continuous analogue exactly, measure the rounding error via a sawtooth sum of fractional parts, reduce the needed bound to a finite counting inequality for multiples of 1/√r, and close that inequality with a parity argument on r−⌊√r⌋² plus direct che
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The central claim is that the sequence (A_{d+2})_{d≥1} is sub-Fibonacci: A_3 = A_4 = 1 and A_{d+2} ≤ A_{d+1} + A_d for d > 2, with the stronger monotonicity A_d ≤ A_{d+1} holding for d > 2. Here A_d counts the finite O-sequences of multiplicity d whose last non-zero value is at least 2. The proof uses the identity A_d = 1 + Σ O(d−ℓ−1, d−1, i, d−j)·O(d−ℓ, i−1, k−1, j) over appropriate ranges, and shows A_{d+1} − A_d is a sum of non-negative remainder terms plus terms bounded below by the corresponding terms of A_d via the inequality O(p,n,k,d) ≥ O(p,n,k,d−1). The same iterative framework yields a generating-function algorithm that computes O_d for 1 ≤ d ≤ 1100, and the resulting data support
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If a triangular number belongs to a D(a)-pair of triangular numbers, then it can be extended to infinitely many distinct D(a)-triples consisting entirely of triangular numbers. The same work also exhibits infinite families of a that admit such pairs and infinite families of a that admit none.
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In encoder-decoder transformers trained on one-step Collatz prediction, the long delay between training-set fit and generalization is caused by limited decoder access to structure the encoder has already learned, not by delayed acquisition of that structure. Early encoder organization of parity and residue appears within a few thousand steps while accuracy remains near chance; transplanting a trained encoder accelerates grokking by 2.75 times, transplanting a trained decoder hurts, and freezing a converged encoder while retraining only the decoder eliminates the plateau and raises final accuracy from 86.1 percent to 97.6 percent. Choice of numeral base further modulates how much local digit
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Theorem 3.1 (Dyadic frequency law): {F(s) : 2^k ≤ s < 2^{k+1}} = {3+ν_2(j) : 1 ≤ j ≤ 2^k} as multisets, equivalently #{s∈B_k : F(s)=r} = 2^{k-r+2} for 3≤r≤k+2 and #{s∈B_k : F(s)=k+3}=1. The paper also proves the dyadic endpoint identity T(2^k)=2^{k+2}-k-2 and the exact defect identity R(n)=2t_1(n+1)-t_1(2n+1)-1.
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The Paradigm Exhaustion Theorem states that every known framework for promoting distributional convergence (almost all orbits descend) to pointwise convergence (all orbits descend) encounters an irreducible structural obstruction when applied to the Syracuse map.
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The central claim is Theorem 1: if s is a Somos sequence of order 6 with nonzero terms, then every 5×5 diamond minor of s×s vanishes, so the diamond rank is at most 4. The new order-7 analogue (Theorem 3) says every 5×5 half-diamond minor vanishes, where half-diamond means one of the two offset families is congruent modulo 4; these bounds also hold generically for twinned pairs (Theorems 2 and 4). The proof works by checking contiguous minors and then using a non-degeneracy lemma to pass to all minors.
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The announced result is a non-definability theorem: for every odd q≥3 and d≥1 with q+d a power of 2, the reachability relation R(x,z) of the generalized Collatz map T_{q,d} — the relation 'z is some iterate of x' — has no first-order definition in the structure ⟨N,+,V_q⟩, where V_q(x) is the largest power of q dividing x. Because first-order definability in this structure coincides with recognition by finite automata reading base-q digits, this is equivalently the statement that no finite automaton can decide reachability. The proof strategy is to show that a definition of R would yield a definition of the set of powers of 2, which is impossible by a classical theorem on automata-recognizabl
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The paper's central claim is that the arbitrary-step transition relation of the generalized Collatz map T_{q,d} is not first-order definable in base-2 Büchi arithmetic. The announced mechanism: definability of that reachability relation would put the exponential set P_q={q^y:y∈N} inside the same logic, where semilinearity forbids exponential growth. The supplied body instead shows that, for each fixed y≥1, the set of pairs (x,C) with C divisible by 2^x−3^y is not semilinear, because its x-fiber is an arithmetic progression with period 2^x−3^y and these periods are unbounded. It also shows every admissible Collatz parity pattern has a unique 2-adic solution, a ghost cycle, that is a genuine p
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On the paper's own terms, the discovery is Theorem B: every map of the form f(n)=f0(n) for odd n, n/2 for even n — in particular the Collatz, Syracuse, and Baker maps — has only finitely many periodic orbits. The route is a 'dictionary': after defining the coarsest topology containing all two-point sets {n,2n}, the Borel structure makes f measurable, periodic orbits are open, every f-invariant probability is supported on a periodic orbit, and entropy vanishes on all of them. Pressure therefore reduces to a supremum of integrals over cycle measures; the author proves that every continuous potential integrable with respect to all invariant probabilities attains its pressure, then invokes Lemma
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For derangement numbers D_n, the paper proves that D_{m+n}/n! equals a double sum over l and k of binomial coefficients, a sign, and a factor k!/l! times D_l. For harmonic numbers H_n, it proves the binomial-weighted identity binom(m+n,m) H_{m+n} = sum_{k=0}^n H_k binom(m+n-k-1,n-k) + H_m binom(m+n,n), and the degenerate version replaces the last binomial's top by m+n-λ and uses degenerate harmonic numbers H_{n,λ}. All three recurrences are obtained by the same manoeuvre: take the known univariate exponential generating function, expand it as a bivariate series in x and y, factor it as a function of x times a function of y/(1-x), expand each factor, and compare coefficients of x^n y^m.
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The central claim is that for n ≥ 3, a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)) takes only the value 1 or prime values, where b(n) obeys b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The paper further states that b(n−3) + n b(n−4) can be replaced by (n−1)! in the gcd without changing the result, yielding a finite continued fraction representation and a connection to the left factorial function. The sequence is claimed to be a denser prime source than Rowland's recurrence, and a symmetry conjecture predicts that each prime (except 5) appears exactly twice at positions related to p.
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On the paper's own terms, the discovery is that the Kurepa factorial is governed by the polynomial F_n(x) = sum_{k=0}^n k! x^k, obtained from the Fubini polynomial by fixing the Stirling number S(n,k) at k=1 (or k=n). Setting x=1 recovers the left factorial !(n+1), and its half r_n = F_n/2 is odd for n≥3. Theorem 16 then claims gcd(F_n,(n+1)!) = gcd(2r_n,2T) = 2·gcd(r_n,T) = 2, using the binary GCD split and the assertion that gcd(r_n,(n+1)!/2)=1. The author presents this as a new equivalence to Kurepa's conjecture and as the capstone of a web of Bell and Dobinski decompositions.
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On the paper's own terms, the central discovery is Theorem 2.2: for every $m,n\ge 0$ and $\lambda\ne 0$, $$\mathrm{Bel}_{n+m,\$\lambda$}(t)=\sum_{k=0}^{m}\sum_{l=0}^{n}(1)_{k,\$\lambda$}\binom{m}{k}_{\$\lambda$}\binom{n}{l}$t^{{k}}$$F^{{(k)}}$_{n-l,\$\lambda$}(-\$\lambda$ t,k-m\$\lambda$)\,\mathrm{Bel}_{l,\$\lambda$}(t),$$ where $\mathrm{Bel}_{n,\lambda}(t)$ are the fully degenerate Bell polynomials, $\binom{m}{k}_{\lambda}$ are degenerate Stirling numbers of the second kind, and $F^{(k)}_{n,\lambda}(x,y)$ are two-variable degenerate Fubini polynomials of order $k$. Theorem 2.3 gives the analogous identity $$F_{n+m,\$\lambda$}(t)=\sum_{k=0}^{m}\sum_{l=0}^{n}k!\binom{m}{k}_{\$\lambda$}\binom{n}{l}$t^{{k}}$$F^{{(k)}}$_{n-l,\$\lambda$}(t,k-m\$\lambda$)\,F_{l,\$\lambda$}(t)$$ for degenerate Fubini polynomials. A first pass with $t=1$ yields the number-level identity of Theorem 2.1. Both polynomial identities are meant to hold as formal power series in $t$, and the paper shows that taking $\lambda\to 0$ recovers the classical Spivey-type recurrences for ordinary Bell polynomials and Fubini polynomials, with the Bell limit reading $\varphi_{n+m}(t)=\sum_{k,l}\binom{m}{k}\binom{n}{l}t^k k^{n-l}\varphi_l(t)$.
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The central claim is Theorem 1.1. For every n≥1 there are unique multivariate polynomials ψ_ν(x_1,...,x_ν) ∈ Z[x_1,...,x_ν] with no constant term, computable recursively, such that for every odd prime p>n, W_p ≡ Σ_{ν=1}^{n} p^{ν-1}/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^n, and equivalently (p-1)! ≡ -1 + Σ_{ν=1}^{n} p^ν/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^{n+1}. Here Q_p(ν) is the ν-th power sum of Fermat quotients. The ψ_ν do not depend on p: once computed, they are universal coefficients for the p-adic development of the Wilson quotient at every order. The proof obtains them from a coefficient-extraction argument in which elementary symmetric polynomials of Fermat quotients are rewritten as power sums
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Starting from the generating function for the unsigned new type degenerate Stirling numbers of the first kind, the paper proves Theorem 2.1: these numbers equal Σ_{m=k}^n λ^{n−m} binom(n,m) [m k]. The main result, Theorem 2.2, states that for n≥k≥1, [n k]^*_{−λ} = Σ_{m=k}^n λ^{m−k} binom(n,m) S1(m,k) E[(S_k)_{n−m,λ}], where S_k is the sum of k independent products of a uniform(0,1) and an exponential(1) random variable. The proof uses the identity E[e^{S_k}_λ(t)] = ((1/λ)log(1+λt))^{-k} log^k(1/(1−(1/λ)log(1+λt))), which converts the defining generating function into an expectation of a degenerate exponential. Letting λ→0 recovers the classical identity [n k] = binom(n,k) E[S_k^{n−k}]. The c
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The discovery is an operational substitution rule: if Σ t^n/n! a_n(x)=A(t)e^{xt}, then a_n(x)=A(∂_x)x^n. Writing the Padé approximant [r|s]A(t)=P_r(t)/Q_s(t) and replacing t by ∂_x gives [r|s]a_n(x)=P_r(∂_x)Q_s(∂_x)^{-1}x^n, an Appell-type approximation of a_n. Expanding the inverse denominator as a formal series turns this into combinations of truncated exponential polynomials; for Hermite amplitudes it yields, for instance, [1|1]He_n(x)=e_n^{(2)}(x,-1/4)-n(n-1)/4 e_{n-2}^{(2)}(x,-1/4), and for [0|2] and [3|2] the expansions are expressed through second-kind Chebyshev polynomials U_r(a,b). The paper further shows the approximated polynomials inherit quasi-monomial operators, so recurrences
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The central claim is Corollary 3.2: for every n ≥ 1, exactly 2^{n−1}(2^n−1) restricted colored base-3 partitions of (3^n−3)/2 satisfy condition (1.1). In particular, when n is prime and 2^n−1 is also prime, that count is a perfect number; conversely, by the Euclid–Euler characterization, every even perfect number is obtained by choosing n equal to a Mersenne-prime exponent. The paper explains the mechanism: the subsequence Q_n obeys a second-order linear recurrence whose characteristic polynomial factors as (t−4)(t−2), and the Binet formula is exactly the Euclid–Euler product. The authors further identify the companion subsequence R_n = S((3^n−1)/2;Z), which gives 2^{n−1}(2^n+1), and develop
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For a fixed ℓ, a multiple almost-Riordan array is the lower-triangular matrix whose column 0 has generating function b(t) and whose remaining columns are tg times cyclic products of ℓ multiplier series f1,...,fℓ. The paper asserts that these arrays close under the multiplication rule (37), with identity (1|1;t,...,t) and inverse (38), forming the multiple almost-Riordan group MaR. The central sequence characterization is Theorem 3.1: every entry is generated by one A-sequence, ℓ Zj-sequences, and one W-sequence, whose generating functions (40)–(43) are explicit combinations of b, g, f1,...,fℓ and the compositional inverse h of h = ℓ√(f1⋯fℓ). The production matrix (44) is assembled from these
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Fix coprime positive integers u,v and set t_n^{(u,v)} = F_{n-2}u + F_{n-1}v. The paper proves that for every n, the pair (a,b) = (t_n, t_{n+1}) has its unique nonnegative integer solution to exactly one of (1.1)/(1.2) given by a closed form (Phi, Psi). Theorems 3.5–3.10 cover the six residue classes of n modulo 6, with the superscript on Phi and Psi recording which equation is used. The regime is chosen by n mod 6, the parities of u and v, and a residue r satisfying v r ≡ ±1 mod u (u odd) or v r ≡ ±1 or ±(u+1) mod 2u (u even). The proof verifies an algebraic identity making the coefficients solve the equation, then checks in tables that they are nonnegative integers. Taking u=v=1 recovers th
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The paper's central claim is that the number s(n) of distinct remainders n mod k for 1 ≤ k ≤ floor(n/2) has an exact linear asymptotic density: s(n) = c·n + O(n/(log n log log n)), where c = Σ_p 1/(p(p+1)) · Π_{q<p}(1 − 1/q) ≈ 0.2296. This confirms, with a rigorous error term, a density that had previously only been conjectured from numerical data. The paper also proves that s(n+1) is never more than one larger than s(n), that arbitrarily large downward jumps occur, and that every downward jump is at most O(log log n) in size. For the iterated remainder sets S_j(n), it establishes density bounds 1/(j+2)! ≤ liminf s_j(n)/n and limsup s_j(n)/n ≤ 1/(j+2), tying the problem to the length of Pier
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There is a well-behaved family of canonical collections for which the question of whether their sumset equals all of Z is equivalent to the universal halting problem for Fractran and is therefore undecidable, via a translation of the covering question into dynamical systems that preserves the undecidability.
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Let $Y$ be a random variable whose moment generating function converges in an interval around $0$. The paper constructs probabilistic Frobenius-Euler polynomials and their degenerate and higher-order variants, and claims that every polynomial in $x$ is a finite linear combination of them. The coefficients in such an expansion are given explicitly through umbral calculus: the sequence of moments of $Y$ is treated as a linear functional, and the formulas express the coefficients in terms of these moments and the Frobenius parameter. The paper illustrates the expansions for several concrete choices of $Y$.
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The central claim is that the generating-function identities for degenerate exponentials, degenerate logarithms, and degenerate polylogarithms can be combined with binomial inversion to produce explicit closed-form evaluations. Specifically, Theorem 2.1 gives H_{n,λ}=Σ_{k=1}^n binom(n,k)(-1)^{k-1} $k^{{-1}}$ binom(λ+k-1,k-1), and Theorem 2.9 gives $H^{{(k)}}$_{n,λ}=Σ_{m=1}^n (-1)^{m-1} $m^{{-k}}$ binom(λ-1,m-1). The paper also introduces $K^{{(m)}}$_{n,λ} by the generating function -(1-t)^{-1} Li_{m,-λ}(-t/(1-t)) and proves that it equals a nested chain sum over ordered indices, an alternating binomial sum, and a Lah-number expansion; for m=1 it reduces to H_{n,λ}. Taken together, these identities assert that the degenerate harmonic family is transparently expressible through elementary finite sums for all positive integers and all real parameters λ for which the degenerate functions are defined.
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The paper's central claim is that a single normalization rule creates a working calculus for degenerate polynomial families. The degenerate Sheffer-type polynomials $T_{n,\lambda}^{(a,b)}(x)$ satisfy the difference rule $T_{n,\lambda}^{(a,b)}(x+1)-T_{n,\lambda}^{(a,b)}(x)=nT_{n-1,\lambda}^{(a-1,b)}(x)$ and the averaging rule $T_{n,\lambda}^{(a,b-1)}(x)=\frac12(T_{n,\lambda}^{(a,b)}(x+1)+T_{n,\lambda}^{(a,b)}(x))$, and these combine into the convolution identities of Theorems 3.10 and 3.11 for higher-order degenerate Bernoulli and Euler polynomials. In the probabilistic half, the paper says that for any $Y$ whose moment generating function exists in a neighborhood of the origin, the family $S^Y_{n,\lambda}(x)$ defined by the normalized generating function satisfies $E[S^Y_{n,\lambda}(x+Y)]=(x)_{n,\lambda}$. For $Y\sim U[0,1]$ the coefficients are finite sums involving Stirling numbers of the first kind and degenerate Bernoulli polynomials, while for $Y\sim \mathrm{Ber}(1/2)$ one gets exactly $S^Y_{n,\lambda}(x)=E_{n,\lambda}(x)$, the degenerate Euler polynomials.
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The paper asserts that for each elliptic curve y^2 - a x y - y = x^3 - b x^2 - c x, the Hankel transform of the sequence u_n with generating function g(x) begins 1, ac-b-c^2, a^2c-a(b+3c^2)+2bc+2c^3-1, ... and 'is a (1, -ac+b+c^2) Somos 4 sequence, which essentially coincides with the elliptic divisibility sequence of the elliptic curve.' If true, every such curve has a lattice path model whose return-count Hankel transform is essentially its elliptic divisibility sequence.
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On the paper's own terms, the discovery is Theorem 1: for every fixed integer $X\ge 10$, $$\int_2^X \$\Delta$^8(x)\,dx = \frac{35C_7-28C_4}{2048\$pi^{8}$}\int_2^X $x^{2}$\,dx + O($X^{{3-1/254+\varepsilon}}$),$$ where $C_4$ and $C_7$ are the explicit eight-variable sums in (14) and (15). These sums run over 8-tuples of natural numbers satisfying an equality of sums of square roots, with weights involving $d(n)$ and powers of the variables. The proof first obtains the analogous statement for the truncated sum $\Sigma_Y(x)$ on a dyadic interval $[H,2H]$, controlling all oscillating contributions; only the two non-oscillating matchings survive to produce the constants. Summing dyadic intervals yields the stated asymptotic, and a short-interval version appears as Theorem 2.
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Theorem 1 states that the n×n Pulsar puzzle has one and only one solution. In that solution, the circled cells, read as a spiral from the center outward, contain the dual of the Pulsar sequence (each term x replaced by n+1−x), and the uncircled cells, also read from the center, contain the Pulsar sequence itself. A stronger property holds as well: in every row and column, every circled digit is larger than every uncircled digit. The proof is by induction, building the n×n solution from the (n−1)×(n−1) solution by adding 1 to the circled entries and completing the removed row according to the symmetric-sum rule of the sequence.
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The central claim is that the fractal dimension of $X_A^{[\alpha,\beta,\gamma,\delta]}$ is completely determined by the transition matrix $A$ and by how the positive integers split into forward orbits of the map $f(\lfloor\alpha k+\beta\rfloor)=\lfloor\gamma k+\delta\rfloor$. Writing $d_i$ for the density of integers whose orbit under $f$ has exactly $i$ points before leaving a specified region, and $d_\infty$ for the density of infinite orbits, the Minkowski dimension equals $$\dim_M $X_A^{{[\alpha,\beta,\gamma,\delta]}}$ = \sum_{i=1}^{\infty} \left[ \frac{1}{(\gamma/\$\alpha$)^{i-1}} d_i + \left(\frac{1}{(\gamma/\$\alpha$)^{i-1}} - \frac{1}{(\gamma/\$\alpha$)^i}\right)\left(\sum_{j>i} d_j + d_\infty\right)\right] \log_m |$A^{{i-1}}$|,$$ for irreducible $A$. For primitive $A$, the Hausdorff dimension equals $$\dim_H $X_A^{{[\alpha,\beta,\gamma,\delta]}}$ = d_1 + \sum_{i=2}^{\infty} d_i \log_m t_{\emptyset,i} + d_\infty \log_m \sum_{i=0}^{m-1} t_i,$$ where $t$ is the unique positive vector with $t_i^{\gamma/\alpha}=\sum_j A(i,j)t_j$. Equality of the two dimensions holds if and only if the row sums of $A$ are all equal. This is an extension: the previously treated cases with integer parameters $(p,a,q,b)$ and $p<q$ are special choices of the real parameters.