claims depot shelf
The Collatz Conjecture
Formal claims (Lean)
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The central claim is that the two PTM logical states $|0_{\mathrm{TM}}^{(N)}\rangle$ and $|1_{\mathrm{TM}}^{(N)}\rangle$, built from the equal-power-sum split of the first $2^N$ Thue–Morse indices, form a protected code subspace in an X-X Ising spin chain. For any product of fewer than $N$ Pauli $\sigma_z$ operators on distinct qubits, the diagonal matrix elements for the two states agree and the
Stated claims
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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 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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We prove that for each m≥1, exactly F(m+1) odd integers in {1,…,2^m} have the property that their orbit under T avoids the residue class 4 mod6 during steps 2,…,m, where F(m+1) is the (m+1)-th Fibonacci number; the proportion decays at rate (ϕ/2)^m, ϕ=(1+√5)/2. The proof uses the directed graph G of Collatz transitions modulo 6 and its unique absorbing strongly connected component G'={1,2,4,5}. Removing vertex 4 from G' yields a subgraph of spectral radius ϕ, against ρ(G')=2; the Fibonacci count follows from this spectral gap.
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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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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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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 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.
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Theorem 1.3 states that under the hypotheses of Theorem 1.2 — $h\neq 0$, and the $h$-th powers of the characteristic roots are distinct and not equal to $1$ — the identity (5) determines exactly one tuple $(P_1,\ldots,P_{m+1})\in\mathbb{C}[x]^{m+1}$. Existence is shown by an explicit induction that builds the polynomials from an invertible matrix $M(d,h)$ and the step-$h$ recurrence coefficients $e_i(h)$. Uniqueness is shown by proving that the only solution of the homogeneous relation $\gamma_1(n)s_{(n+1)h+r}+\cdots+\gamma_m(n)s_{(n+m)h+r}+\gamma_{m+1}(n)=0$ for all $n\in\mathbb{N}$ is $\gamma_1=\cdots=\gamma_{m+1}=0$; the proof passes through the Binet form, groups roots by modulus, and uses Lemma 2.6 to rule out every nontrivial oscillatory term, ending with an invertible Vandermonde system (34) that forces each $\gamma_i$ to vanish.
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The central claim is a conditional structural discovery: assuming both the 3x+1 conjecture and the Weak Residue Conjecture $\mathrm{Res}(N)\le 2$, the paper derives six identities, beginning with $O(N)=\lfloor \log_6 2\,D(N)-\log_6 N\rfloor$ and $E(N)=\lceil \log_6 3\,D(N)+\log_6 N\rceil$, with four analogous formulas obtained by using $D$, $O$, or $E$ as the known quantity. The derivation starts from $1\le \mathrm{Res}(N)\le 2$, takes logarithms, and produces a width-one interval that contains exactly one integer, so the floor or ceiling is forced. The paper further proves, under only the 3x+1 assumption, that $\mathrm{Res}(N)<O(N)^{1/9}$ for $O(N)\ge 20$, and hence $\mathrm{Res}(N)<2$ whenever $O(N)\le 512$; this supports the Weak Residue Conjecture without completing its proof.
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The central claim is Theorem 3.1: for every $m\ge 2$, no positive integers $n,k,r$ satisfy $L_m^{n+k}+L_m^n=L_r$. In the $n=1$ case the proof shows that $L_m$ must divide the difference $L_r-L_m$ and uses a factorization depending on $r-m \bmod 4$ to force either a factor of $5$ dividing a Lucas number (impossible) or a primitive prime divisor of a large Lucas factor dividing a smaller one (also impossible), leaving only two finite pairs to check. For $n\ge 2$, the proof uses an exact divisibility property to bound $m$ and $n$, then an explicit lower bound for linear forms in logarithms to bound $k$, then a continued-fraction criterion to push $k$ down to at most $8$, and finally an explicit computer search. The theorem is presented as complete: after the reductions only a finite range remains, and the authors report checking it exhaustively.
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The central claim is that a simple hybrid rule—$n$ belongs to the Primender sequence iff $n$ is prime or $n \bmod 10^k$ is prime for some $k\ge1$—generates a sequence with stable, checkable structure and that this structure makes a good hidden-rule test for LLMs. The paper reports two structural facts: the gap $\Delta$ between consecutive Primender terms is never larger than 5, and whenever $\mathrm{PE}_n - \mathrm{LP}_n = 1$ (the $n$-th term is one more than the largest prime ≤ it), the gap from the previous term is 1. These facts are verified computationally on the first 100,000 and 1,000,000 terms. On the benchmark, the paper reports that only ChatGPT o3 inferred the rule from the first 100 terms, judged the hypothesis correctly, and generated the next 100,000 terms with a multiset error of 5.16%, while the other eight models failed at rule inference, produced high-error lists, or returned no usable list.
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On the paper's own terms, Theorem 3.1 is the central claim: the sequence defined by the recurrence $a(n+1)=n-a^{(m)}(n)+a^{(m+1)}(n)$, $a(1)=1$, is uniquely $a_{\mathrm{sol}}(n)=n-h(n)$, with $h(n)=\left\lfloor \frac{m-2+\sqrt{(m-2)^2+8m(n-1)}}{2m}\right\rfloor$. The uniqueness follows by induction once the key identity $h(n+1)-1=h(a_{\mathrm{sol}}^{(m)}(n))$ is established for all $n\ge1$. The paper proves that identity by splitting $n$ into boundary cases and interior cases, using lemmas that pin down the values of $h$ after iterating $a_{\mathrm{sol}}$ at the boundary indices $T_k^*$. It also reports that $a(n)$ is slowly growing, that it increments exactly at the non-special integers, and that for $m=1,2,3,4$ the solution matches known tabulated sequences connected with triangular, square, pentagonal, and hexagonal numbers.
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The paper defines deformed bivariate q-Appell polynomials P^{(α)}_{n,q}(x,y;u) via (A_q(t))^α e_q(tx) e_q(ty,u) = Σ P^{(α)}_{n,q}(x,y;u) t^n/[n]_q!, and claims that this class enjoys explicit coefficient formulas, q-derivative relations, addition properties, operator representations, and Mehler and Rogers type identities. If correct, this provides a unified class of bivariate deformed q-Appell polynomials that specializes to known families for special u.
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The central claim is that the inverse Riordan array $\left(\frac{1}{1+x},\frac{x}{(1+x)^r}\right)^{-1}$, called the pre-Fuss-Catalan-Riordan array, has entries $\tau_{n,k}=\frac{rk+1}{(r-1)n+k+1}\binom{rn}{n-k}$, and that the Fuss-Catalan-Riordan array $(g_r(x),xg_r(x))$ factors as $(g_r(x),xg_r(x))=(g_r(x),xg_r(x)^r)\cdot\left(1,\frac{x}{(1+x)^{r-1}}\right)$, where $g_r(x)=1+xg_r(x)^r$ is the generating function of the $r$-th Fuss-Catalan numbers. This factorization expresses each Fuss-Catalan triangle as the product of the d-orthogonal moment array, whose production matrix has A-sequence $(1+x)^r$ and Z-sequence $(1+x)^{r-1}$, with a binomial-type Riordan array. The rectified Fuss-Catalan matrix $(g_r(x),g_r(x))$ is recovered by multiplying the pre-Fuss-Catalan array on the right by the transpose of the binomial matrix, giving the explicit sum formula for its entries. The same banded production matrices tie the construction to constant-coefficient d-orthogonal polynomial recurrences and to lattice paths with step sets $\{(1,1),(2-r,1-r)\}$ and $\{(0,1),(1,1-r)\}$.
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On the paper's own terms, the central discovery is Theorem 1: for any discrete A⊂R>0 with limsup_{x→∞} (1/log x) ∑_{α∈A∩[1,x]} 1/α > 0 and any ε>0, there exist distinct α,β∈A and n∈N with |nα−β|<ε. Iterating the theorem after deleting each found pair gives infinitely many such pairs. The proof proceeds by contradiction, assuming that |nα−β|≥1 for all distinct α,β and all n∈N. From that assumption the paper derives lim_{T→∞} (1/T)∑_{α∈A∩[1,T]}1 = 0, which contradicts the divergence of the reciprocal sums. This density-zero conclusion is exactly what resolves the problem; no quantitative rate such as (1.6) is obtained, and the authors explicitly describe the proof as soft.
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The paper's central discovery is that the exponents of any fixed prime p across the positive integers—the p-adic valuation sequence—form a fractal sequence, and that for p=2 the same sequence, drawn as a path that turns v2(n) quarter-turns at step n, produces the Lévy Dragon. The sieve that reaches this conclusion starts from a single 0 and repeatedly expands it by making p−1 copies, appending them, and increasing the final term by 1; Theorem 6 shows this reproduces prime factorizations without division. Lemma 12 exhibits a simple decimation rule—take every (p+1)-st term—that locates a copy of the whole sequence inside itself, and Lemma 13 shows no periodic block generates it. Theorem 15 proves that the terms at indices divisible by 8 exactly match an algorithm for the Lévy Dragon's turn sequence, with the intervening terms forming small 'T' detours. In the appendix, the same construction is carried for the odd part of n, whose values modulo 4 match the Heighway Dragon's turns.
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The central claim is Theorem 4: if $n=m=p$ is an odd prime and $\delta$ is the order of $2$ modulo $p$, then the only possible values of $\mathrm{Per}(\mathbf{u})$ on $\mathbb{Z}_p^p$ are $1$, $\delta$, and $p\delta$. The zero tuple alone has period $1$; a tuple $(x,\ldots,x)$ with $x\neq 0$ has period $\delta$; every other tuple has period $P_p(p)=p\delta$. The paper also establishes Theorem 3 for $n=3$: for odd prime $m$, every tuple $\mathbf{u}=(x_1,x_2,x_3)$ whose entries are not all equal and whose sum is not $0$ modulo $m$ has period $P_m(3)$, while nonzero sum-zero tuples have period at most $6$ by Lemma 2.
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The paper's central claim is that for the listed $(n,m)$, the eventual Ducci cycle is closed under the cyclic shift $H$, witnessed by fixed power identities on the cycle subgroup $K(\mathbb{Z}_m^n)$. For $n=3$, the intended identities are $D^2=H$ when $m$ is a power of $2$; $D^{m-1}=H^2$ on all of $\mathbb{Z}_m^3$ when $m\equiv 5\pmod 6$ is prime; and $D^{p-1}=H^2$ on the cycle when $m=2^\ell p$ with $p\equiv 5\pmod 6$ prime. For even $n$ with $m\equiv -1\pmod n$ prime, the identity is $D^{m-1}=H^{-1}$ on the cycle. The proof obtains these by reducing to the basic Ducci sequence—every tuple is a linear combination of cyclic shifts of $(0,\ldots,0,1)$, and $D$ commutes with $H$—and then verifying the identities with the binomial coefficients that appear in the iterates of $(0,\ldots,0,1)$.
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We identify paradoxical sequences of finite length in the Collatz iteration that exceed their initial term contrary to expectation from the odd-term proportion when iterating beyond the stopping time. This non-typical behavior is closely related to the Collatz conjecture. It most likely occurs finitely many times, thus lending support to Terras' conjecture.
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The paper establishes a small-gap theorem: there is a constant c (the proof gives c = 7 eventually) such that for every n one has min_{1≤k≤n} a_{k+1}/a_k ≤ 1 + c log n / n. It also improves the best growth bound to a_n ≤ 1.454^n for all sufficiently large n. The growth bound is obtained by showing that Eggleton's recurrence cannot be applied twice in a row, leaving three possible linear recurrences whose products are controlled through submultiplicativity of operator norms and a finite check over all admissible words of length 15. The small-gap result is obtained by a counting argument that compares candidate sums a_n + a_i with alternative representations a_j + a_k.
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The paper's central claim is the Second Fundamental Theorem of the Quanta Prime Sequence (Theorem 25): for any nonzero point $(\alpha,\beta)$ and any natural number $n$, the ratio $\Omega_0(\lfloor n/2\rfloor|\alpha,\beta|n)$ over $(n-1)(n-2)\cdots(n-\lfloor n/2\rfloor)$ equals $\Psi(\alpha,\beta,n)$ and is an integer. From this single identity the author derives new quotient representations for Mersenne numbers, Fermat numbers, Lucas numbers, the Fibonacci-Lucas oscillating sequence, Chebyshev polynomials, and Dickson polynomials, together with the divisibility statement $p_{k+1}\mid \Omega_0(p_k|\alpha,\beta|2p_k)$ for the $k$th prime $p_k$. The paper also presents this as the proof of the previously unproved Theorem 9 from the prior paper [3], recasting Mersenne primality in terms of divisibility of $\Omega$-quotients.
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The central discovery is that the coefficient table of $p_n$ is rigid in $n$. Writing $p_n(x)=\sum_{k=0}^{2^n}c_{n,2k}x^{2k}$, the paper proves (Theorem 2.5) that for every $n$ with $1\le k\le 2^n$, $c_{n,2k}=\sum_{j=1}^k a_{j,k}2^{2jn}$, with $c_{n,0}=2$, where the $a_{j,k}$ are rational numbers defined recursively and independent of $n$. The proof begins from the reciprocal identity $x^{2^{n+1}}p_n(x+x^{-1})=x^{2^{n+2}}+1$, which turns coefficient extraction into an upper-triangular linear system (Theorem 2.1); then the recursion $p_n=p_{n-1}^2-2$ separates the powers of $2^{2n}$. For the diagonal, Proposition 3.5 shows $a_{k,k}=\sum_{T\in\mathcal{T}_k}\prod_{v\in V_T}b_v^{\delta_{v,T}}$, where $b_1=-1$, $b_v=2^{-2}(2^{2(v-1)}-1)^{-1}$ for $v>1$, and $\mathcal{T}_k$ is the set of labeled ordered trees whose cardinality is the $(k-1)$-st Catalan number.
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Denote $s(n,N)=\sum_{k=1}^n k^N$. Starting from $s(n,0)=n+1$, the paper proves that for $N\ge 1$, $$s(n,N)=\frac{(n+1)^{N+1}-\sum_{j=0}^{N-1}\binom{N+1}{j}s(n,j)}{N+1}.$$ This is equation (3) of the paper. The proof expands $(k+1)^{N+1}-k^{N+1}$ by the binomial theorem, sums over $k$, and isolates $s(n,N)$. The paper also notes that $s(n,N)$ is a polynomial in $n$ of degree $N+1$ with leading term $n^{N+1}/(N+1)$, and reports an empirical factorization checked by computer algebra for $N\le 100$: for odd $N\ge 3$ the polynomial is divisible by $n^2(n+1)^2$, and for even $N\ge 2$ it is divisible by $n(n+1)(2n+1)$.
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The central claim is that the two PTM logical states $|0_{\mathrm{TM}}^{(N)}\rangle$ and $|1_{\mathrm{TM}}^{(N)}\rangle$, built from the equal-power-sum split of the first $2^N$ Thue–Morse indices, form a protected code subspace in an X-X Ising spin chain. For any product of fewer than $N$ Pauli $\sigma_z$ operators on distinct qubits, the diagonal matrix elements for the two states agree and the off-diagonal elements vanish (Property 1.2.2); the paper identifies this as the Knill–Laflamme condition, so up to $(N-1)/2$ single-qubit phase-flip errors are detectable and, given ancillas, correctable. Every X-X rotation $e^{i\theta \sigma_x^{(k)}\sigma_x^{(j)}}$ leaves each PTM state unchanged up to the same global phase, making the encoded subspace transparent to the X-X Ising Hamiltonian. The two states are also exchanged by $S_x$, the Hadamard transform maps them from simple computational-basis superpositions, and the same Prouhet–Tarry–Escott identity extends the construction to qudits.
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The paper's central result is a pair of equivalent integral characterizations. Theorem 1 says that for formal series $g(t)=\sum \mu_{n,\nu}t^n/\gamma_{n,\nu}$ and $f(t)=\sum a_n t^n/\gamma_{n,\nu}$ with nonzero leading coefficients, a sequence $\{s_{n,\nu}\}$ is the Sheffer-Dunkl sequence for $(g,f)$ if and only if some function $\alpha_\nu$ of bounded variation on $\mathbb{R}$ has moments $\mu_{n,\nu}=\int x^n\,d\alpha_\nu(x)$ and satisfies $\int L_f^r s_{n,\nu}(x)\,d\alpha_\nu(x)=\gamma_{n,\nu}\delta_{n,r}$. Theorem 3 says $\{s_{n,\nu}\}$ is Sheffer-Dunkl if and only if some bounded-variation $\beta_\nu$ with nonzero zeroth moment represents $s_{n,\nu}(x)=\int \tau_t(p_{n,\nu})(x)\,d\beta_\nu(t)$, where $p_{n,\nu}$ are the associated Dunkl polynomials for $f$. In the first form the moments of $\alpha_\nu$ are the coefficients of $g(t)$; in the second, the moments of $\beta_\nu$ are the coefficients of $1/g(t)$. The proofs run through the generating identity $1/(g(\bar f(t)))\,E_\nu(x\bar f(t))=\sum s_{n,\nu}(x)t^n/\gamma_{n,\nu}$ and the degree-lowering action of $L_f$.
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On its own terms, the paper establishes that a Collatz trajectory can be compressed by treating every even number as a member of a branch that starts at an odd root and extends by powers of 2. The algorithm extracts the largest power of 2 dividing the current even number with the bitwise trick $n \wedge -n$, divides it out to reach the odd root, and then applies $3n+1$ to move to the parent branch. Repeating this until the base branch ($2^m$) is reached gives a loop count of exactly $2k+1$ for $k$ sub-branches. The paper verifies the computed stopping times against a brute-force implementation for the first billion numbers, and reports a consistent 28.0% reduction in iterations compared to the bitwise brute-force algorithm of Venkatesulu and Parameswari [3] and the code-word algorithm of Ren et al. [2], including for numbers as large as $2^{100000}-1$.
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The integer patterns inferred from the arborescence graph constructed from iterations of g(x) provide new insights into proving the validity of the Collatz conjecture.