claims depot shelf
Goldbach's Conjecture
Formal claims (Lean)
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On the paper's own terms, the discovery is a finite reduction of the two-prime-sum problem to a single numerical condition. For $x=12n+4$, the degree in $G(x/2)$ is $n-f(n)+1$ or $n-f(n)$ according as $x-3$ is prime, where $f(n)$ is the size of the union of the divisibility events $A_p$ for primes $3<p<\sqrt{x}$; analogous formulas are proved for $x=12n+k$, $k=0,2,4,6,8,10$. The paper proves $\pro
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The central claim of the paper is square-averaged cancellation for the binary sums: $\sum_{n \le N} |G_2(n)/n|^2 \ll_A N/\log^A N$. From this statement the paper derives the log-power bound for all $k \ge 3$ by a short induction, the density-zero statement for the exceptional set in Theorem 2, and the equidistribution of Liouville sign patterns in Theorem 3. The full limit $\lim_{N\to\infty} |G_2(
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The paper's central claim is that every integer $N\ge2$ can be written as $N=a+b$ with $a,b$ positive integers and $\Omega(ab)\le33$. The proof works with the auxiliary sequence $a_n=n(N-n)$, halved when $N$ is odd, and keeps only indices with $3\nmid a_n$; the prime 3 is then excluded from the sifting set. This pre-sieving removes the extremal $w=3$ case from the dimension condition, allowing a s
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The central claim (Theorem 1.1) is a lower bound for $v(n)$, the number of ordered prime tuples satisfying the mixed-power equation, namely $v(n) \gg n^{39/37}/\log^7 n$ for all sufficiently large odd $n$. The proof defines the exponential sum $G(\alpha)=g_2(\alpha)^2g_3(\alpha)g_4(\alpha)g_5(\alpha)g_6(\alpha)g_8(\alpha)$, with $g_5$ summed only over primes $p \le n^{\gamma/5}$, and shows that $\
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The paper's central claim is Theorem 1.1: for sufficiently large positive integer $T$, the minimal degree $D(T)=\min\{\deg\tau\,:\, \operatorname{tr}(\tau)\le -T,\ \tau\text{ a Salem number}\}$ satisfies $$\frac{$2T^{2}$}{16\log(2T)+11}\le D(T)\le 2(1+o(1))$T^{2}$\log T.$$ Equivalently, the degree of the smallest Salem number with trace at most $-T$ is roughly $T^2$, up to logarithmic factors. The
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The paper establishes two new results. First, Proposition 7.5: for R = X^ϑ with 0 < ϑ < 4/9, the smoothed Goldbach count r_φ(N) satisfies Σ_{N≤X} |r_φ(N) − N S(N) − M(N;R) − Z(N;R)|² ≪ (X^{3−ϑ} + X^{13/5})(log X)^5, where M and Z collect, explicitly, the contribution of every zero of every Dirichlet L-function of conductor at most R. Second, Theorem 8.2: if for some A > 5/2 and δ ∈ (0,1) the spars
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Under GRH and the bound J_1(T)≪T, the character-averaged weighted sum of Λ(n)χ(n)μ(m)f((n+m)/N) (and its logarithmic counterpart) is O_ε(N^{2−ε}E(f″)) for every Sobolev weight f∈W^{2,1} supported in [0,β), for all levels of distribution θ<1; the same size holds for Hölder–Zygmund weights of order δ with a θ that depends on δ but is always at least 1/2−2ε.
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For every fixed integer k≥4 the sum of r2(n)^k over n≤x is asymptotic in order of magnitude to x(log x)^{2^{k-1}-2k-1}. The upper bound for k=4 loses the previous log-log-log factor, the third-moment error reaches the conjecturally optimal O(x/(log x)^3), and all lower bounds hold unconditionally.
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There exists a positive constant δ such that, for all sufficiently large N, at least δ N integers n ≤ N admit a representation n = m + a^a with Ω(m) ≤ 2 and a a natural number.
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The paper proves that under GRH the Linnik–Goldbach problem can be solved with K=6, meaning every sufficiently large even integer is p1+p2+2^a1+...+2^a6, improving the previous K=7. Unconditionally, it proves that the lower density of numbers representable as p+2^a is at least 0.12532, so at least a quarter of odd numbers are of this form. Both improvements follow from replacing the classical siev
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Theorem 1.3: For fixed q ≥ 2 and k ≥ 2, under GRH for Dirichlet L-functions modulo q, the average G_{q,k}(N) satisfies G_{q,k}(N) = Σ_k(q)/φ(q) · G_{1,k}(N) + O(N^{1/k} log^2 N log q / φ(q)), where Σ_k(q) = Σ_{χ^k = χ_0} χ(−1). The main term is therefore a constant multiple of the unrestricted average G_{1,k}(N), and the constant is an explicit character sum that can vanish. When Σ_k(q) = 0, the a
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The central claim is Theorem 1.2: for every N≥2 there exist positive integers a and b with N=a+b and Ω(ab)≤40; Theorem 3.2 refines this to Ω(ab)≤39 when N is even. The proof works by showing that the sifted set S(A,z) is non-empty for z=N^{1/20}, where A={n(N−n)} for even N (and half-values for odd N). Any surviving element has all prime factors at least z, and because the elements are <N^2, this
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Theorems 1.1 and 1.2 assert that for h = X^θ with 2/15+ε < θ < 0.99, all but O_ε(X exp(-c_ε (log X)^{1/4})) integers x ∈ [X,2X] satisfy R(x) ≍ h and S(x) ≍ h, where R(x) counts representations of n ∈ (x,x+h] as p+a (p prime, a in a lacunary set A_λ(X) of size ≍ log X) and S(x) counts distinct such n. The lower bound R(x) ≫ h comes from a quoted almost-all primes-in-short-intervals asymptotic (rest
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The central claim is Theorem 1: for fixed integers $j\ge k\ge 2$ and any $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that $\sum_{n=N+1}^{N+H} R_{\phi,j}(n) = (\gamma_k^j/\gamma_{k,j})\,H N^{(j-k)/k} + O(H N^{(j-k)/k} A(N;-C))$, uniformly for $N^{1-13/(15k)+\varepsilon}<H<N^{1-\varepsilon}$, where $\gamma_k=\Gamma(1+1/k)$, $\gamma_{k,j}=\Gamma(j/k)$, and $R_{\phi,j}(n)$ is the weighted
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In the paper's own terms, the discovery is a realizability barrier: a proof in HA of statements of the form ∀x(φ(x)⇒∃yψ(x,y)) must yield a total, closed realizer t such that for every x, ψ(x,t(x)) holds. For primality, φ(n)=PrimeΠ(n) is classified as Π1 and compositeness as ∃ with Σ1 witness; the paper asserts there is no total primitive-recursive functional, and no closed term in any strongly nor
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In the paper's own terms, the central claim is that for every large x the number of natural numbers n ≡ 4 (mod 6) with n ≤ x that cannot be expressed as a sum of two Chen primes is O(x^{1−δ}) for some absolute δ > 0. The authors construct a non-negative model for the Chen primes: a sieve-based weight that is everywhere no larger than the indicator of a Chen prime, yet captures enough of the set th
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The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of le
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The central discovery is that the $\tfrac{5}{8}$ density threshold for the three-prime representation theorem is best possible: there exists an infinite set of primes $Q$ with lower relative density exactly $\tfrac{5}{8}$ and upper relative density $1$ for which infinitely many odd integers $2n+1$ are not sums of three elements of $Q$. The proof interleaves blocks of all primes with blocks of prim
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The paper's central claim is the conjecture that every integer $n > 23$ can be written as a sum of at most five terms each of the form $p^k$, where $p$ is prime and $k \ge 2$. The author states this as a conjecture, not a theorem, and presents computational support: all integers up to $10^7$ were checked exhaustively, and large numbers up to $10^{10}$ were checked by sampling, with no exception fo
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On the paper's own terms, the central discovery is a uniform bound for the sums $S_\mu(\mathcal{D};M)$ defined in (1.5): under the hypotheses of Theorem 2.1, $S_\mu(\mathcal{D};M) \ll_{C,\varepsilon,\delta} (A+B)N(\log N)^{-C}$ for every $C>0$, uniformly in $M \in \mathbb{Z}$. The bound holds for arbitrary complex weights $u_n,v_n$ with sup norm at most $1$, for injective functions $f$ and $g$ on
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On the paper's own terms, the discovery is that the sparse set $\{f_{k^2}:k\in\mathbb N_0\}$ behaves like a two-term additive basis for the primes in a strong density sense. The paper constructs a modulus $M$ and a residue class $n_0\pmod M$ with the property that any representation of such an $n$ forces the prime to lie in a single residue class and both square indices to lie in two prescribed re
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The paper's central claim is that a length-$N$ CAZAC sequence set can be built for any $N$ by choosing primes with $N=Q_1+Q_2$ (even) or $N=Q_1+Q_2+Q_3$ (odd), concatenating a length-$Q_1$ block with length-$Q_2$ (and $Q_3$) blocks, and repeating the shorter blocks across the set of $Q_1$ cyclic shifts. Lemma 1 gives the exact inner product: two extended sequences are orthogonal unless they overla
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The central discovery is that both representation problems can be disposed of by a single congruence construction. Two covering systems, one of 33 congruences for Fibonacci indices and one of 15 congruences for Lucas indices, force the index of any hypothetical representation into a residue class modulo a listed prime. Because the target number and the forced Fibonacci or Lucas term are congruent
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Theorem 1.1 states the central equivalence. If $\zeta(\sigma+it)\neq 0$ for $\sigma>1-\eta(\log|t|)$, then the smooth weighted error $\Delta(x)=\sum_n(\Lambda(n)-1)e^{-n/x}$ satisfies $\Delta(x)\ll x\exp(-(1-\epsilon)\omega_\eta(x))$, where $\omega_\eta(x)=\inf_{t\geq 1}(\eta(t)\log x+\log t)$. Conversely, if $\Delta(x)\ll x\exp(-(1-\epsilon)\varpi(x))$ with $\varpi(x)=\min_{u\geq 0}(\eta(u)\log x
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The paper's central assertion is that failure of the Goldbach conjecture is governed by the density of primes, not by their specific identities. For an even number $2n$, define $A_n$ as the set of distances $n-q$ for elements $q\in Q$ at or below $n$, and $B_n$ as the distances $q-n$ for elements $q\in Q$ between $n$ and $2n$. A representation $2n=q_1+q_2$ exists exactly when $A_n\cap B_n$ is none
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The paper's central computational discovery is that the set $A=\{a: a^2+1\text{ is prime}\}$ has the Goldbach property through $a=2.5\times10^{14}$: every element of $A$ with $1<a\le 2.5\times10^{14}$ is the sum of two elements of $A$. This is established by a segmented three-stage sieve that produces the complete list of primes $m^2+1<6.25\times10^{28}$, a list occupying more than 30 terabytes. T
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The central claim is Theorem 1: for any finite shift set $H$ and any small prime set $P$, the limit $\kappa_H^P=\lim_{x\to\infty}\frac{1}{x}\sum_{n\le x}\prod_{h\in H}\lambda_P(n+h)$ exists and equals $\prod_{p\in P}(1-2\eta_p^H)$, where $\eta_p^H=d/(p+1)$ whenever $p$ divides none of the differences $h_i-h_j$, and exceptional primes have explicitly computable constants. The proof works through th
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The paper's central claim is Theorem 1.2: if some fixed $c_2>0$ has the property that $L(s,\chi)$ never vanishes in $\Re(s)>1-c_2$ for any Dirichlet character $\chi$, then for $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ and base $g$ sufficiently large, almost all even numbers $m \in [X,X+H]^*$ are Goldbach numbers; equivalently, the set of even exceptions has size $o(|[X,X+H]^*|)$. The proof
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Under GRH, for every large prime N and each sign pattern (eta1,eta2), the number of n<N with lambda(n)=eta1 and lambda(N-n)=eta2 is at least c*N*exp(-C*(log log N)^6).
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Under GRH, every sufficiently large even N is a sum a+b=N with lambda(a)=lambda(b)=-1.
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The paper's central discovery is Theorem 1.5: for $\varepsilon\in(0,1/3]$, $A,K\ge 1$, with $z_0=\exp(25000A^3K(\log 2A)^2/\varepsilon)$, $M$ divisible by $P(z_0)$ with all prime factors below $z_0$, and $N\ge (Mz_0)^{1+\varepsilon^{-1}}$, every subset $\mathcal{P}^*$ of primes in $[\sqrt N,N]$ with $|\mathcal{P}^*|\ge N/(K\log N)$ can be written as $1_{\mathcal{P}^*}=f^\flat/(V(z_0)\log N)+f^\sha
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The central discovery is Theorem 3.6. If $S$ is an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group, then every $f\in S[G]$ with $|\operatorname{supp}(f)|>1$ is a sum of at most two irreducibles if and only if $\mathscr{A}_+(S)=S^\times$, and this condition is also equivalent to the weaker-looking bound at most $n$ irreducibles for any fixed $n>1$. N
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Stated on its own terms, the paper's central claim is the analytical aVGC: for every pair of finite positive integers $a\ge b\ge 0$ there is a finite positive integer $L(a,b)=L(b,a)$ such that for all integers $n>L(a,b)$ one can write $2n=p^{a+1}(x)+p^{b+1}(y)$ with the two summands distinct, where $p^{k}(x)$ denotes the $x$-th prime after applying the prime-indexing function $k$ times, so $0$-pri
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The central assertion is Theorem 1.1: if $H<X$ and $\log H/\log_2 X\to\infty$, then $\sum_{h\le H}|\sum_{p\le X}\mu(p+h)|=o(H\pi(X))$; and if $H=X^\theta$ for $\theta\in(0,1)$, the same sum is $\ll_{\theta,\delta}H\pi(X)/(\log X)^{1/3-\delta}$ for every $\delta>0$. Thus the folklore conjecture that $\sum_{p\le X}\mu(p+h)=o(\pi(X))$ holds for almost all shifts $h\le H$, with at most $O_A(H(\log X)^
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The central claim is that, for $s\ge 12$, the weighted prime solution count $R(P)$ for the system has the asymptotic form $R(P)\sim C P^{s-5}$ with $C>0$ whenever the system has a nontrivial real solution and, for every prime $p$, the local system has a solution with all coordinates nonzero modulo $p$. The constant is uniform over all admissible choices of the integer coefficients $u_i,v_i$. The p
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The central discovery is an asymptotic rigidity theorem for prime-generated semigroups. Writing $e_n$ for the embedding dimension and $\pi$ for the prime-counting function, the paper proves $\pi(u_n)\sim 3n$, $e_n\sim 2n$, and $u_n\sim 3p_n$. The engine is a strong three-prime theorem with almost equal summands: every sufficiently large odd $N$ is $q_1+q_2+q_3$ with $\lvert q_i - N/3\rvert\le N^\t
Stated claims
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The central claim, Theorem 1, is: for every fixed $\phi\in\mathbb{Z}[x]$ of degree $k\ge 2$ with positive leading coefficient, and every $\varepsilon>0$, the count $E_{k,\phi}(N)$ of indices $n\le N$ for which $2\phi(n)$ is not a sum of two primes satisfies $E_{k,\phi}(N)\ll_{\phi,\varepsilon} N^{\alpha_k+\varepsilon}$, with $\alpha_2=1/2$, $\alpha_3=5/8$, $\alpha_4=3/4$, and $\alpha_k=1-1/(k+1)$ for $k>4$. The paper is explicit that the proof is conditional: it needs $\theta<11/12$, where $\theta$ is the supremum of the real parts of nontrivial zeros of all primitive Dirichlet $L$-functions, and it imports this hypothesis from an external source in the stronger form $\theta\le 7/8$ (its Theorem A, quoted in Section 2). Given that input, two quoted theorems do the analytic work: a minor-arc fourth-moment bound for the weighted prime sum, $\int_{\mathfrak m}|S_w(\alpha)|^4\,d\alpha\ll X^{5/2}(\log X)^{100}$, and a major-arc asymptotic relating the representation integral to the singular series $S(m)$ times a weight convolution of size $\gg X$. The new contribution is the polynomial side: matching the prime fourth moment against mean-value estimates for exponential sums over arbitrary subsets of polynomial sequences, the proof shows the exceptional set for $2\phi(n)$ has size at most $N^{(k+2b)/q+\varepsilon}$, and the three families of subset moment estimates — an elementary divisor argument for degree 2, an eighth-moment discrete-restriction bound for degrees 3 and 4, and Vinogradov mean-value estimates for $k>4$ — convert that master bound into the exponents of Theorem 1.
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The central claim is Theorem 1.1: the ordered Goldbach count $G(2n)$ on the antidiagonal $I_{2n}$ is an $\mathcal{O}(n/\ln^{2} n)$ function of $n$. The proof proceeds by induction, partitioning the $(n-2)$ ordered odd pairs that sum to $2n$ into the prime-prime class $G(2n)$, the composite-composite class $D(2n)$, and the mixed classes $H(2n)$ and $K(2n)$, which agree by symmetry of the antidiagonal. From the identity $G(2n)+H(2n)=\pi(2n-3)$ together with the prime number theorem expansion $\pi(x)=x/\ln x+x/\ln^{2}x+2x/\ln^{3}x+\cdots$, the proof fixes the leading and second-order terms of $H(2n-2)$; positivity of $G(2n-2)$ combined with the induction hypothesis forces the second coefficient $\alpha$ to be strictly below $2$. A sixteen-case analysis of the prime and composite configurations of a pair $(x,y)$ and its shift $(x+2,y-2)$ under the maps $\psi\colon I_{2n}\to I_{2n+2}$ and $\tau\colon I_{2n}\to I_{2n-2}$ then yields $H(2n+2)=H(2n-2)+2R_1-2R_3$, where the equalities $R_1=R_2$ and $R_3=R_4$ come from symmetry of the antidiagonals. Whichever of the surviving sizes $R_3$ has, the second coefficient of $G(2n+2)$ is strictly positive, so every sufficiently large even number has a Goldbach representation and the binary conjecture follows for all even integers greater than four; the corollaries give the ternary statement for odd integers and the bounds $D(2n)=O(n)$ and $H(2n)=K(2n)=O(n/\ln n)$.
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On its own terms, the paper's central claim is that the two quotients attached to Goldring's $w$-function are simultaneously as rich and as sparse as possible on each of the classes $B_3$ and $C_3$: the finite limit-point sets of $R(n)$ and $S(n)$ are exactly $[0,1]$ and $[0,\infty)$, and every nonempty open interval $I \subset (0,1)$ or $J \subset (0,\infty)$ contains values with counts $\gg_I X^{2/3}(\log X)^{-3}$ on $B_3$ and $\gg_I X(\log X)^{-6}$ on $C_3$ (with $w(n) \in C_3$ also imposed). At the same time, for every fixed $\alpha > 0$ the set of $n \le X$ lying in $A_3$, $B_3$ or $C_3$ with $R(n) > \alpha$ has size $o(S(X))$, so $D_{A_3}(\alpha) = 1$ for all $0 < \alpha \le 1$. On $B_3$ the paper describes the joint limit set of $(R(n), S(n))$ exactly as $([0,1] \times \{0\}) \cup (\{0\} \times [0,\infty)) \cup \bigcup_{m \ge 1} \Gamma_m$, where $\Gamma_m$ is the curve traced by $(r_m(u), g_m(u))$ for $u > 0$, with $r_m(u) = \max\{u, (u+1)/(2m)\}/\max\{u,1\}$ and $g_m(u) = (u+1)^2/(4m^2u)$; in particular the section at first coordinate $1/2$ is $\{(1/2,0)\} \cup \{(1/2, 9/(8m^2)) : m \ge 2\}$, so $(1/2, 1)$ is not a joint limit even though $1/2$ and $1$ are limits of the separate quotients.
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On the paper's own terms, the central discovery is one exact identity plus two asymptotic laws. The identity (Eqs. 14 and 17) reads $P(2n) = |G(2n)|/(2n+1)$, equivalently $|G(2n)| = (2n+1)\mathrm{Tr}[(\Pi_P \otimes \Pi_P)\rho_{2n}]$: once the beam-splitter transmissivity is averaged over its uniform range, the probability of detecting a prime pair is exactly proportional to the number of ordered Goldbach representations of $2n$, with no combinatorial weight left on any individual pair. The asymptotic laws govern phase estimation under the Hamiltonian $H = a_1^\dagger a_1 - a_2^\dagger a_2$: for fixed transmissivity $T$, the beam-splitter Goldbach state has quantum Fisher information $I \sim 32T(1-T)n$, giving $\Delta\varphi \sim 1/\sqrt{n}$ (standard quantum limit), whereas the equally weighted Goldbach state has $I \sim (16/3)n^2$, giving $\Delta\varphi \sim 1/n$ (Heisenberg limit). Both laws are derived under the refined Hardy–Littlewood density for weighted Goldbach sums and, for the beam-splitter state, under the explicit assumption that $G(2n)$ is nonempty.
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The proved part is Corollary 2.4: $24m+1$ is prime exactly when $m\notin\bigcup_{i=1}^8 S_i$, where $S_1=\{x+y+24xy:x,y\in\mathbb{Z}^+\}$ and $S_2,\dots,S_8$ are the images of the seven polynomials $1+5(x+y)+24xy$, $2+7(x+y)+24xy$, $5+11(x+y)+24xy$, $7+13(x+y)+24xy$, $12+17(x+y)+24xy$, $15+19(x+y)+24xy$, and $22+23(x+y)+24xy$. These families come from the factor identities $(24x+1)(24y+1)$, $(24x+5)(24y+5)$, ..., $(24x+23)(24y+23)$. The claimed discovery is the Section 3 Claim that $\mathbb{Z}^+$ equals $S_1\cup\cdots\cup S_{56}$, where $S_9,\dots,S_{41}$ are wild-solution families and $S_{42},\dots,S_{56}$ are tame-solution families imported from the authors' earlier work. For each solution set, membership of $m$ is accompanied by an explicit positive-integer solution to the Erdős-Straus equation for $n=24m+1$; for $S_1,\dots,S_8$, membership means $24m+1$ factors. The paper proves the primality criterion and explicitly states that it failed to find a logic proof for the covering claim, presenting instead a numerical test to $m\le 10^{11}$.
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The claim on the paper's own terms is Theorem 1, $E_2(N)\ll_\varepsilon N^{19/24+\varepsilon}$, where $E_2(N)$ counts $m\le N$ such that $2m^2$ is not a sum of two primes. With $N=\sqrt{X/2}$ and $m$ ranging over $[X/2,X]$, the circle method splits the weighted representation count $R(2m^2)=\sum_{p+p'=2m^2}\log p\,\log p'$ into major-arc and minor-arc parts $R_1+R_2$. The major arcs are evaluated through the explicit formula of Theorem A (cited to [Pin2023]), which writes $R_1(m)$ as the singular series $S(m)=2C_0\prod_{p\mid m}(1+1/(p-2))$ plus a sum over a set $E(H,P_0,X)$ of generalized exceptional zeros of primitive Dirichlet $L$-functions with conductor up to $P_0=X^{4/9-\varepsilon}$; Theorem B, cited to [OAI2026a], states that no Dirichlet $L$-function has a zero in $\Re s>7/8$, which empties $E$ and turns the formula into the clean asymptotic $R_1(2m^2)=(1+o(1))S(2m^2)\cdot 2m^2>2m^2\ge N^2$ (Theorem C). On the minor arcs, Theorem D (cited to [MPR2026]) bounds the prime exponential sum pointwise, and because $\sqrt{P_0}>X^{5/24}$ the argument obtains $\sup_{\alpha\in\mathfrak{m}}|S(\alpha)|\le X^{19/24+\varepsilon}$; with the fourth moment $\int_0^1|f(\alpha)|^4\,d\alpha\ll X\log^3 X$ (via the classical two-square divisor bound $a_n\le 4d(n)$) and the sign trick of [BKW2001], Hölder's inequality gives $\sum_{m\le N}|R_2(2m^2)|\ll X\cdot N^{19/24+\varepsilon}$. Since the major-arc term $R_1(2m^2)>2m^2$ outweighs this averaged minor-arc total, (3.12) concludes $R(2m^2)>0$ apart from an exceptional set of size $N^{19/24+o(1)}$, which is the theorem.
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The central result, Theorem 1.1, asserts that for 1 < c < 5179/3909 and any large E, the Diophantine inequality |p_1^c+p_2^c+p_3^c+p_4^c - N| < (\log N)^{-E} is solvable in primes p_i such that each p_i+2 has at most \lfloor \frac{31250}{5179-3909c}\rfloor prime factors. The proof constructs a positive lower bound for a weighted sum over such primes, showing it exceeds zero for all sufficiently large N. The weight uses a smooth cutoff function and Fourier inversion to approximate the inequality condition, and the positivity follows from a combination of exponential sum estimates and a linear sieve inequality that compares the contributions of the 'minus' and 'plus' sieve weights.
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The central claim is Theorem 1.2: for an infinite strictly increasing sequence $\{a_k\}$ with $a_k\mid a_{k+1}$ for every $k$, the condition $\sup_{k\ge1}\sum_{d\mid a_k}\frac1d=\infty$ forces the representation set to have asymptotic density zero. The authors obtain this by converting the infinite condition into divergence of the reciprocal sum of the prime divisors appearing across the whole sequence, then choosing a finite set $F$ of odd primes whose reciprocal sum $H$ is arbitrarily large. Because the sequence is a divisibility chain, the product $Q$ of those primes eventually divides every $a_k$, so a residue-class weight $T$ evaluated on a represented integer is the same as $T(p^2+q^2)$. The proof bounds the integers where $T$ is large by $O(X/H)$, bounds the prime-pair sums with small $T$ by $O(\pi(\sqrt{X})^2/H)$, and uses $A(X)=O(\log X)$ to obtain $\overline{d}(S(A))\le \frac1H(4+40/(\log 2)^2)$; letting $H\to\infty$ gives density zero.
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The central claim, stated as Theorems 1 and 2, is that under $(Q_\Theta)$ — every nontrivial zero of every primitive Dirichlet $L$-function satisfies $\Re\rho \le \Theta$ for a fixed $1/2 \le \Theta < 11/12$ — one has $E_+(N) \ll_\Theta N^{1/2}(\log N)^{100}$ and $E_-(N) \ll_\Theta N^{1/2}(\log N)^{100}$. Here $E_+(N)$ counts even integers $n$ with $2 < n \le N$ that are not sums of two primes, and $E_-(N)$ counts positive even $h \le N$ that are not differences of two primes. The bridge between the two theorems is the fourth-moment bound $\int_{\mathfrak{m}(X^\kappa)} |S_w(\alpha)|^4\,d\alpha \ll_{w,\Theta} X^{5/2}(\log X)^{100}$ for the smoothed prime exponential sum $S_w(\alpha) = \sum_p (\log p)\,w(p/X)\,e(p\alpha)$ on the minor arcs, with $\kappa = \kappa(\Theta) \in (4/9, 9/20)$. By additive orthogonality, the minor-arc Fourier coefficients of $S_w(\alpha)^2$ and $|S_w(\alpha)|^2$ have $\ell^2$-sums exactly equal to that fourth moment; on the major arcs the corresponding coefficients are at least a constant times $X$ for every even $n \in [X/2, X]$ and every even $h \in [2, X/2]$, so a missing representation contributes a full $X^2$ to the $\ell^2$-sum and forces the number of exceptions down to the square-root scale. Structurally, the proof isolates the principal character's contribution, then for the nonprincipal part associates each large value of the complete prime polynomial with a nearby zero of a primitive Dirichlet $L$-function, and balances the zero-density saving against the energy of the large-value set, with an exact four-stage identity capping the terminal decomposition at eight factors and fixing the logarithmic exponent.
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The central discovery is a topological reformulation of the strong Goldbach conjecture: the conjecture is true if and only if the intersection of all singleton closures in the Goldbach relational space (E6, τ_G) equals {6}. This is equivalent to the space failing the weak separation property wR0 and being connected. The authors further show that compactness of the space is equivalent to the finiteness of the set of even integers that are not sums of two primes. The construction relies on a multivalued map that sends an even integer m to the set {p+3 : p is a prime with m−p prime}, and then considers the Alexandroff topology whose open sets are those that are 'predecessor-invariant' under this relation.
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The central discovery is that, under H_{7/8}, the exceptional set $E_\square(N)$ satisfies $E_\square(N) \ll_\varepsilon N^{11/20+\varepsilon}$ for every $\varepsilon>0$. The improvement comes from a new uniform estimate (Lemma 6) for weighted zero‑counts of families $C_d(T,K)$ of primitive characters obtained by twisting a fixed quadratic character $\lambda_d$ with characters of varying conductors $t$. The argument controls the fourth moment of $L(1/2+iu,\chi)$ over such families using an approximate functional equation and the large sieve, then applies zero‑detection to absorb all bands of zeros with real part at least $129/167$. The remaining bands and the minor arcs give the final exponent $11/20$.
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The core discovery is Theorem 1.2: for the counting function α(x) of the set of integers not representable as a prime plus a Zumkeller number, we have α(x) ≪ x/(log log x)^A, where A = log_2($2π^{2}$/($2π^{2}$-3)) ≈ 0.2378. A direct consequence is that the natural density of this set is zero, confirming Conjecture 1.1 of Somu, Kukla, and Tran.
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The paper proves Theorem 1: For every positive integer r, there exist infinitely many positive integers n such that L(n) - K(n) ≥ r, where L(n) is the maximum number of distinct primes summing to n and K(n) is the number of parts in the unique partition with the largest product. This affirmatively settles Guy's problem.
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The central claim is that for every sufficiently large even integer N there exist primes p_1, p_2 and positive integers ν_1,…,ν_K with K ≤ 6 (unconditionally) or K ≤ 5 (under GRH) such that N = p_1 + p_2 + $2^{{ν_1}}$ + ⋯ + $2^{{ν_K}}$. More precisely, the paper establishes explicit lower bounds r''_6(N) ≥ 1/1000 · N $L^{6}$ and r''_5(N) ≥ 1/1000 · N $L^{5}$, where L ≈ log_2 N. These bounds are obtained via a reflection argument that converts the representation problem into a second‑moment estimate for a signed counting function f(n) = r'_3(n) − a L r'_2(n) − b $L^{2}$ r'_1(n), whose positive part g(n) is then used in a Cauchy‑Schwarz inequality. The second moment ∫|S(α) F(G(α))|^2 dα is split into major and minor arcs; on the major arcs it is asymptotically 2N $L^{6}$[(1−a−b)^2 + v^T A v] plus a term involving $b^{2}$ $L^{5}$ R_1(0), while on the minor arcs the contribution is bounded by a constant times the minor‑arc integral of |S G|^2 plus an error that is o(N $L^{6}$) thanks to a large‑deviation set E_λ(L) where the normalised dyadic sum z_L(α) is far from the region D_λ. Choosing a = 1/4, b = 1/16 gives the unconditional bound; under GRH an additional weight τ = 2 and a second signed function f_2(n) lead to the five‑power bound.
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The paper establishes two quantitative statements about additive decompositions of the primes. Theorem 1.1: if A,B ⊆ N have at least two elements and the symmetric difference (A+B)△P is finite, then √x/log x ≪_{A,B} A(x), B(x) ≪_{A,B} √x as x→∞. Theorem 1.3: for k ≥ 3 and sets A_1,…,A_k ⊆ N each with at least two elements, the symmetric difference ((A_1+⋯+A_k)△P)(x) ≫ $x^{{1−3/(2k)}}$/(log x)^{5/2−3/(2k)}; in particular for k=3 the bound is √x/(log x)^2. These are the first power-law lower bounds for the ternary case and remove iterated-log factors from the binary case.
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The central discovery is that the exponent in the Goldbach exceptional set can be conditionally reduced to $7/12$ (Theorem~1) and, under the uniform zero bound $H_\theta$ with $\theta\le2/3$, to the even smaller exponent $g(\theta)=\theta(5-4\theta)/(2(2-\theta))$ (Theorem~3). The proof of Theorem~1 combines Pintz's explicit major‑arc formula with the exponential‑sum estimate of Maynard, Pandey, and Radziwiłł to handle the minor arcs, while Theorem~3 uses a classical fixed‑modulus zero‑density estimate in place of the new exponential‑sum bound. The key insight is that once the major arcs contribute a positive main term (guaranteed by the zero‑free hypothesis), the minor‑arc estimate becomes the limiting factor, and the choice of zero‑density estimate determines the final exponent.
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Theorem~1.1: Let $r_1,\dots,r_t$ be positive integers with $r_1^{-1}+\cdots+r_t^{-1}=1$. Let $r(n)$ count the representations $n=p+2^{k_1^{r_1}}+\cdots+2^{k_t^{r_t}}$ with $p$ an odd prime and positive integers $k_i$. Then for every positive integer $m$, the set $\{ \text{odd } n : r(n) \ge m \}$ has positive lower asymptotic density. This is sharp: if the reciprocal sum is less than $1$, the density is zero.
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The central claim is that five recurrence summands, mixed in any way between the two sequences, join two primes to represent every large integer. Precisely, there exists $N_0>0$ such that for every $N\ge N_0$ and every $u\in\{1,2,3,4\}$ there is a representation $N=p_1+p_2+\sum_{i=1}^u F_{r_i}+\sum_{j=1}^{5-u}L_{s_j}$ with $p_1,p_2$ primes and all $r_i,s_j$ positive; repeated primes and repeated recurrence values are allowed, and ordered representations count distinct index positions. The proof separates the recurrence sequences by parity: the index class $j\equiv2\pmod3$ carries the even Fibonacci values, the complementary classes the odd Fibonacci values, multiples of three the even Lucas values, and the remaining classes the odd Lucas values. For each $u$ and each parity of $N$, the integral $\int_{\mathbb{T}}S(\alpha)^2W_{u,\nu}(\alpha)e(-N\alpha)\,d\alpha$ counts weighted representations, where $W_{u,\nu}$ is a fixed nonnegative polynomial in four normalized exponential sums whose monomials each contain $u$ Fibonacci factors and $5-u$ Lucas factors. The major arcs contribute at least $1.99426N$ (and $1.99405N$ for the even $u=4$ case, $1.9939N$ for the odd), while the minor arcs contribute $o(N)$: for $u=1,2,3$ by a pointwise prime-sum bound together with a large-deviation estimate of exponent $11667/20000$, and for $u=4$ by the fourth-moment bound of exponent $2.5401$ combined with exceptional-set decay exponents above $0.561$. The argument closes because each major-arc constant exceeds the corresponding minor-arc loss by margins ranging from about $2.5\times10^{-5}$ up to about $4.6\times10^{-2}$.
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Theorem 1.3: assume GRH for Dirichlet L-functions with characters mod q. Then for every j at least 3, G_{q,j}(N) = (C_{q,j}/phi(q)) * G_{1,j}(N) + O_q(N^{j-1} log^2 N), where C_{q,j} = phi_{j-1}(q)/phi(q)^{j-1} is positive unless both j-1 and q are even. If correct, the average over multiples of q of the j-prime von-Mangoldt-weighted representation count is fully determined by the constant C_{q,j} and the known unweighted average G_{1,j}(N) ~ N^j/j!.
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The central claim, in the paper's own words, is that “many problems already contain within themselves all the structural information needed to solve them,” and that “the real difficulty is not that computational power is insufficient, but that the structure of the problem has not been identified.” On this view, the successes the paper surveys—a formalized proof of a weakened Liouville-variant Goldbach conjecture, the three-dimensional Kakeya resolution, Noether's theorem, and the author's geophysical inversion methods—share one logic: deconstruct, identify structure, exploit structure to build the solution directly, without iterative search. The paper is explicit that structure-driven methodology does not replace objective-driven methodology but complements it: objective-driven is a general search strategy, structure-driven a custom direct strategy, and when structure is missing the structure-driven approach degenerates into the objective-driven one. The significance claimed is not only efficiency but a shift in the first question a problem-solver asks, from “how to solve faster” to “what is the structure of the problem.”
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The paper proves that every integer n with 14 < n < $10^{26}$ can be written as p + $x^{2}$ + $y^{2}$, where p is an odd prime and x,y are positive integers (Theorem 1.3). Similarly, every integer n with 308 < n < $10^{16}$ can be written as p + $x^{3}$ + $y^{3}$, with p an odd prime and x,y positive integers (Corollary 3.5). These are the best verified bounds for these two conjectures. The proof relies on two main computational inputs: (1) explicit lists of all non-square and non-cube integers below $10^{14}$ and $10^{12}$, respectively, that cannot be written as a prime plus a single square or cube (the exceptional sets E2 and E3), and (2) a bootstrapping theorem that exploits these lists to cover the much larger ranges for two powers.
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The paper proves that for every integer n > 24 with 997# ∤ n, there exists a prime p < n and a non-square-free integer r > 0 such that n = p + r. The core of the proof is the least-missing-prime criterion: if n is not divisible by the smallest prime q that does not divide n, then using the fact that q−# divides n, and comparing with the maximum least prime in residue classes modulo $q^{2}$ (denoted M(q)), one can find a suitable p. For q ≤ 13 the required n > M(q) is ensured by a direct verification up to 3617; for 17 ≤ q ≤ 997, a computation shows M(q) < q−# ≤ n. The value 997 is the computational cutoff.
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The central discovery is that the arithmetic conditions m/n ≈ r and p_m + p_n = □ (or |p_m − p_n| = □) are simultaneously satisfiable with high density. For every fixed open interval I ⊂ ℝ_{>0}, there exists a constant c_I > 0 such that for any L > 0, as N → ∞, all but O(N/(log N)^L) indices n ≤ N admit at least c_I √n/(log(2n))^{3/2} choices of m with m/n ∈ I and p_m + p_n a square; the same holds for the difference and for the roles of numerator and denominator. It follows that the sets R_Σ = {m/n : p_m + p_n is a square} and R_Δ = {m/n : |p_m − p_n| is a square} are both dense in ℝ_{>0}.
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The central claim is that, under the digit non-resonance condition (every prime divisor of $\gcd\{a-a': a,a'\in\mathcal{D}\}$ divides the base $b$) and $\alpha_{b,D}$-admissibility of the restricted-digit Fourier transform $\hat{C}_k$, the von Mangoldt-weighted counts have the exact shapes $T_k^{(2)} = \bigl(1 - \tfrac{1}{(b-1)|\mathcal{D}|}\sum_{a\in\mathcal{D}} a\bigr)\, S_{b,D}^{(2)}\, X|\mathcal{C}_k| + O_{b,D,A}\bigl(|\mathcal{C}_k|X(\log X)^{-A}\bigr)$ and $T_k^{(3)} = \kappa_{b,D}\, S_{b,D}^{(3)}\, X|\mathcal{C}_k| + O_{b,D,A}\bigl(|\mathcal{C}_k|X(\log X)^{-A}\bigr)$, where $S_{b,D}^{(2)}$ and $S_{b,D}^{(3)}$ are finite averages of local congruence factors over the least significant digit and $\kappa_{b,D}$ is the limiting average of the archimedean weight $(1-2m/b^j)_+$ over the digit set. The leading constants are positive exactly when $b$ is odd or $\mathcal{D}$ contains an even digit (pairs) and when $\min\mathcal{D} < (b-1)/2$ together with $\mathcal{D}\cap\gcd(b,6)\mathbb{Z}\neq\emptyset$ (triples). In the positive cases, after removing proper prime powers and the diagonal, genuine primes realize $\gg X|\mathcal{C}_k|(\log X)^{-2}$ pairs and $\gg X|\mathcal{C}_k|(\log X)^{-3}$ three-term progressions with restricted-digit (common) differences. The new input for triples is a three-point minor-arc lemma: for a minor outer frequency $\alpha$, at least one of the three prime frequencies $\beta$, $\alpha-2\beta$, $\beta-\alpha$ in the kernel $\hat{H}_\Lambda(\alpha) = N^{-1}\sum_\beta \hat\Lambda_X(\beta)\hat\Lambda_X(\alpha-2\beta)\hat\Lambda_X(\beta-\alpha)$ decays like $(X^{-1/5}+(DB)^{-1/3+\epsilon})X(\log X)^4$, which forces the stricter threshold $1/6$.
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On the paper's own terms, the discovery is that the method for representing large integers as two primes plus a bounded number of sparse-sequence terms extends from first-order lacunary sequences to the second-order Fibonacci and Lucas recurrences. The Fibonacci numbers are $F_0=0,F_1=1,F_{n+1}=F_n+F_{n-1}$, and the Lucas numbers obey the same recurrence with $L_0=2,L_1=1$. The proof defines a prime sum $S(\alpha)$ over primes in $(N^{0.99},N]$, recurrence sums $T_F,T_L$ over indices up to $J=\lfloor\vartheta\log N/\log\varphi\rfloor$ with $\vartheta=1999/2000$, and representation integrals such as $R_e(N)=\int_0^1 S(\alpha)^2A(\alpha)^2B(\alpha)^4e(-N\alpha)\,d\alpha$, where $A$ and $B$ collect the even and odd Fibonacci terms by index modulo $3$. Positivity of the integer coefficients in these integrals means that a positive value of $R_e(N)$ is exactly a representation. The paper proves that the major arcs contribute more than the minor arcs by a positive margin in each of the three cases.
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On the paper's own terms, the discovery is a finite reduction of the two-prime-sum problem to a single numerical condition. For $x=12n+4$, the degree in $G(x/2)$ is $n-f(n)+1$ or $n-f(n)$ according as $x-3$ is prime, where $f(n)$ is the size of the union of the divisibility events $A_p$ for primes $3<p<\sqrt{x}$; analogous formulas are proved for $x=12n+k$, $k=0,2,4,6,8,10$. The paper proves $\prod_{2<p\le y}(1-2/p)\sim e^{-0.183407}/(\log y)^2$, which converts the independent-events approximation into the closed form $\eta(x)$ and shows it is $1.26095$ times half the classical conjectured count. Then, using the definition $\lambda(A,B)=(P(B)-P(A\cap B))/(P(B)-P(A)P(B))$ and $\lambda_0=\max_i\lambda(\bigcup_{j<i}A_j,A_i)$, it proves that whenever $\lambda_0<2.5$ the union probability satisfies $T(n)<1-2/n$ for $x=12n+4$, $n>447$, so the vertex $x$ has more than two neighbours and is a sum of two odd primes; Theorem 6.10 states the same for every sufficiently large even $x$.
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The central claim of the paper is square-averaged cancellation for the binary sums: $\sum_{n \le N} |G_2(n)/n|^2 \ll_A N/\log^A N$. From this statement the paper derives the log-power bound for all $k \ge 3$ by a short induction, the density-zero statement for the exceptional set in Theorem 2, and the equidistribution of Liouville sign patterns in Theorem 3. The full limit $\lim_{N\to\infty} |G_2(N)|/N = 0$ is not claimed; what is proved is that any limit point of the sequence must be $0$, so the conjecture would follow if the ratio converges. The proof is not conditional on any unproved hypothesis once the classical uniform exponential-sum bound for $\lambda$ is accepted.
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The paper's central claim is that every integer $N\ge2$ can be written as $N=a+b$ with $a,b$ positive integers and $\Omega(ab)\le33$. The proof works with the auxiliary sequence $a_n=n(N-n)$, halved when $N$ is odd, and keeps only indices with $3\nmid a_n$; the prime 3 is then excluded from the sifting set. This pre-sieving removes the extremal $w=3$ case from the dimension condition, allowing a smaller sieve constant $K$. The remainder terms caused by pre-sieving are handled by the inclusion--exclusion identity $r_d=\rho_d-\rho_{3d}+g(d)\rho_3$. The argument then splits into three ranges: the initial range $2\le N\le N_0$ is covered by explicit maximal-prime-gap data, the intermediate range by finite verification of a uniform lower bound, and the large range $N\ge N_1=10^{132}$ by analytic estimates for the main-term factor and the accumulated error term.
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The central claim (Theorem 1.1) is a lower bound for $v(n)$, the number of ordered prime tuples satisfying the mixed-power equation, namely $v(n) \gg n^{39/37}/\log^7 n$ for all sufficiently large odd $n$. The proof defines the exponential sum $G(\alpha)=g_2(\alpha)^2g_3(\alpha)g_4(\alpha)g_5(\alpha)g_6(\alpha)g_8(\alpha)$, with $g_5$ summed only over primes $p \le n^{\gamma/5}$, and shows that $\int_0^1 G(\alpha)e(-\alpha n)\,d\alpha \gg n^{39/37}$. The unit interval is split into major arcs, the set $\mathfrak n$ where the cubic exponential sum is small, and the remaining minor arcs; the major arcs contribute a positive constant times $n^{39/37}$, the small-cubic-sum region contributes $O(n^{39/37-2/8325+\varepsilon})$ via new mixed mean-value estimates, and the remaining region contributes $O(n^{39/37}\log^{-1} n)$ via rational approximation and prime Weyl-sum bounds. This is the first all-prime result for the equation with an eighth power among the variables.
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Conjecture (S) — every even n ≥ 6 can be written as p + t with p prime and t a twin member — sits strictly between the Goldbach and Dubner conjectures. The paper's central discovery is twofold. First, (S) is a single elementary statement that implies both the Goldbach conjecture and the twin prime conjecture (Theorem 1). Second, any proof of (S) would have to contend with a mod-3 orientation rigidity: for even n not divisible by 3, every twin member appearing in a Goldbach partition of n is a lower member (t+2 prime) when n ≡ 1 mod 3, and an upper member (t−2 prime) when n ≡ 2 mod 3, with the only exceptions being the primes 3 and 5. The paper verifies (S) exhaustively to 10^12, computing th
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The paper's central claim is Theorem 1.1: for sufficiently large positive integer $T$, the minimal degree $D(T)=\min\{\deg\tau\,:\, \operatorname{tr}(\tau)\le -T,\ \tau\text{ a Salem number}\}$ satisfies $$\frac{$2T^{2}$}{16\log(2T)+11}\le D(T)\le 2(1+o(1))$T^{2}$\log T.$$ Equivalently, the degree of the smallest Salem number with trace at most $-T$ is roughly $T^2$, up to logarithmic factors. The proof has two independent halves: an upper bound obtained by constructing, for each large odd $n$, a Salem number with trace $-\lfloor n/2\rfloor+o(n)$ and degree at most $n(n+2)(\log n+\log\log n)$, and a lower bound obtained by showing that any Salem number of degree $2d$ with trace $-T$ must satisfy $T^2\le 8d\log(2d)+11d$. The paper also proves Theorem 1.3, that for every odd $n\ge10^4$ there is a constant $d_0(n)$ such that Salem numbers of trace $-\lfloor n/2\rfloor$ occur in every even degree $2d$ with $d\ge d_0(n)$.
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The paper establishes two new results. First, Proposition 7.5: for R = X^ϑ with 0 < ϑ < 4/9, the smoothed Goldbach count r_φ(N) satisfies Σ_{N≤X} |r_φ(N) − N S(N) − M(N;R) − Z(N;R)|² ≪ (X^{3−ϑ} + X^{13/5})(log X)^5, where M and Z collect, explicitly, the contribution of every zero of every Dirichlet L-function of conductor at most R. Second, Theorem 8.2: if for some A > 5/2 and δ ∈ (0,1) the sparse Hardy–Littlewood bounds δS(N)N ≤ r_2(N) ≤ (2−δ)S(N)N hold for all but at most X^{3/5} even multiples N of r̃ in [X/2,X], with X = r̃^A, then no primitive real character χ̃ mod r̃ has a real zero β̃ > 1 − c/log X. The proof of the second result runs by contradiction: if such a zero existed, the Deu
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Under GRH and the bound J_1(T)≪T, the character-averaged weighted sum of Λ(n)χ(n)μ(m)f((n+m)/N) (and its logarithmic counterpart) is O_ε(N^{2−ε}E(f″)) for every Sobolev weight f∈W^{2,1} supported in [0,β), for all levels of distribution θ<1; the same size holds for Hölder–Zygmund weights of order δ with a θ that depends on δ but is always at least 1/2−2ε.
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For every fixed integer k≥4 the sum of r2(n)^k over n≤x is asymptotic in order of magnitude to x(log x)^{2^{k-1}-2k-1}. The upper bound for k=4 loses the previous log-log-log factor, the third-moment error reaches the conjecturally optimal O(x/(log x)^3), and all lower bounds hold unconditionally.
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There exists a positive constant δ such that, for all sufficiently large N, at least δ N integers n ≤ N admit a representation n = m + a^a with Ω(m) ≤ 2 and a a natural number.
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We prove unconditionally that Proposition (1+1.9) is true. Thus every sufficiently large even integer N can be written as N = p + r q, where r ≤ q^{0.9}, r is 1 or prime, and p, q are primes. Assuming the Elliott-Halberstam Conjecture the exponent 1.9 can be replaced by 1.4. The same method yields Proposition (1-1.75) for the twin-prime side unconditionally and (1-1.4) under Elliott-Halberstam.
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Let N be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of N as the sum of three primes, one of which is smaller than a given U. By inserting the currently best zero-density estimate for Dirichlet L-functions, we may unconditionally take U = N^{4/49} exp(log^{2/3 + ε} N) for any ε > 0. If we assume the Generalized Riemann Hypothesis instead, we may take U = log^{4 + ε} N.
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The paper proves that under GRH the Linnik–Goldbach problem can be solved with K=6, meaning every sufficiently large even integer is p1+p2+2^a1+...+2^a6, improving the previous K=7. Unconditionally, it proves that the lower density of numbers representable as p+2^a is at least 0.12532, so at least a quarter of odd numbers are of this form. Both improvements follow from replacing the classical sieve constant C1=8 in the upper bound for counts of prime pairs with the recent C1=6.7814, extended to counts with fixed difference and fixed residue class.
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Theorem 1.3: For fixed q ≥ 2 and k ≥ 2, under GRH for Dirichlet L-functions modulo q, the average G_{q,k}(N) satisfies G_{q,k}(N) = Σ_k(q)/φ(q) · G_{1,k}(N) + O(N^{1/k} log^2 N log q / φ(q)), where Σ_k(q) = Σ_{χ^k = χ_0} χ(−1). The main term is therefore a constant multiple of the unrestricted average G_{1,k}(N), and the constant is an explicit character sum that can vanish. When Σ_k(q) = 0, the average over multiples of q is of smaller order than the unrestricted average, illustrating that k-th powers are not uniformly distributed among residue classes modulo q.
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The central claim is Theorem 1.2: for every N≥2 there exist positive integers a and b with N=a+b and Ω(ab)≤40; Theorem 3.2 refines this to Ω(ab)≤39 when N is even. The proof works by showing that the sifted set S(A,z) is non-empty for z=N^{1/20}, where A={n(N−n)} for even N (and half-values for odd N). Any surviving element has all prime factors at least z, and because the elements are <N^2, this keeps the total number of prime factors of ab below 21, giving the required bound. The large-N case combines the sieve bound with a lower bound for V(z), an upper bound for the remainder via Rankin's trick, and explicit prime estimates; the small and medium cases are verified by computation.
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Theorems 1.1 and 1.2 assert that for h = X^θ with 2/15+ε < θ < 0.99, all but O_ε(X exp(-c_ε (log X)^{1/4})) integers x ∈ [X,2X] satisfy R(x) ≍ h and S(x) ≍ h, where R(x) counts representations of n ∈ (x,x+h] as p+a (p prime, a in a lacunary set A_λ(X) of size ≍ log X) and S(x) counts distinct such n. The lower bound R(x) ≫ h comes from a quoted almost-all primes-in-short-intervals asymptotic (restricting to a ≤ h/2); the upper bound on the second moment Q(x) ≪ h comes from an upper-bound sieve for prime pairs plus a structural estimate on small prime factors in differences a_1-a_2; Cauchy–Schwarz then gives S(x) ≥ R(x)²/Q(x) ≫ h.
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The central claim is Theorem 1: for fixed integers $j\ge k\ge 2$ and any $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that $\sum_{n=N+1}^{N+H} R_{\phi,j}(n) = (\gamma_k^j/\gamma_{k,j})\,H N^{(j-k)/k} + O(H N^{(j-k)/k} A(N;-C))$, uniformly for $N^{1-13/(15k)+\varepsilon}<H<N^{1-\varepsilon}$, where $\gamma_k=\Gamma(1+1/k)$, $\gamma_{k,j}=\Gamma(j/k)$, and $R_{\phi,j}(n)$ is the weighted count of representations by $j$ prime-power values of $\phi$. The paper proves this by writing the short-interval sum as an integral of the $j$-th power of the smoothed exponential sum against a Dirichlet kernel, isolating a major arc around zero, and controlling the two remaining pieces by $L^2$ estimates. It also removes the monic assumption: if $\mathrm{lead}(\phi)=a_k$, the same asymptotic holds with the constant multiplied by $a_k^{-j/k}$.
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In the paper's own terms, the discovery is a realizability barrier: a proof in HA of statements of the form ∀x(φ(x)⇒∃yψ(x,y)) must yield a total, closed realizer t such that for every x, ψ(x,t(x)) holds. For primality, φ(n)=PrimeΠ(n) is classified as Π1 and compositeness as ∃ with Σ1 witness; the paper asserts there is no total primitive-recursive functional, and no closed term in any strongly normalizing typed λ-calculus adequate for HA, that uniformly transforms PrimeΠ evidence into such witnesses. Consequently the constructive Goldbach principle GC*Π has no total Σ1 constructor in HA, so even a classical proof of Goldbach would not give a constructive proof. The paper translates this into
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In the paper's own terms, the central claim is that for every large x the number of natural numbers n ≡ 4 (mod 6) with n ≤ x that cannot be expressed as a sum of two Chen primes is O(x^{1−δ}) for some absolute δ > 0. The authors construct a non-negative model for the Chen primes: a sieve-based weight that is everywhere no larger than the indicator of a Chen prime, yet captures enough of the set that the weighted count of representations of n as a sum of two such primes has a positive main term for every admissible n outside a power-saving exceptional set. This model is assembled with a carefully chosen sifting strategy whose error terms are controlled by a power-saving variant of the Bombier
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The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of length three still occur infinitely often. The paper further states a binary Goldbach-type result for sums of two primes from the same thin Bohr set.
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The central result is Theorem 2: "There is an infinite set Q of primes with upper relative prime density 1 and lower relative prime density 5/8 such that there are infinitely many odd integers 2n+1 that cannot be represented as q1+q2+q3 = 2n+1 with qi in Q (1 <= i <= 3)." If true, this shows Shao's Proposition 2, which guarantees representation when the lower relative density exceeds 5/8, is sharp.
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The paper's central claim is the conjecture that every integer $n > 23$ can be written as a sum of at most five terms each of the form $p^k$, where $p$ is prime and $k \ge 2$. The author states this as a conjecture, not a theorem, and presents computational support: all integers up to $10^7$ were checked exhaustively, and large numbers up to $10^{10}$ were checked by sampling, with no exception found. The significance claimed is that prime powers, despite being sparse and having large gaps, can represent all integers efficiently.
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The paper's central claim is that the Goldbach Conjecture can be interpreted as an informational consistency condition for the set of prime numbers. The argument rests on an empirically proposed Informational Economy Principle: given a difference between primes, the pair realizing that difference tends to have the minimal sum, with overwhelmingly high probability. The author draws an analogy to least-action principles in physics and to fundamental physical limits of computation, treating this observed tendency as a law-like economy rather than a finite-sample accident. Within this heuristic framework, the truth of the Goldbach Conjecture is the normal, expected state of the prime system, while a falsehood would constitute a violation of the economy and thus an extreme informational anomaly. The paper does not claim a proof; it proposes that the difficulty of the conjecture lies in the decoupling of mathematical abstraction from physical information constraints.
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On the paper's own terms, the central discovery is a uniform bound for the sums $S_\mu(\mathcal{D};M)$ defined in (1.5): under the hypotheses of Theorem 2.1, $S_\mu(\mathcal{D};M) \ll_{C,\varepsilon,\delta} (A+B)N(\log N)^{-C}$ for every $C>0$, uniformly in $M \in \mathbb{Z}$. The bound holds for arbitrary complex weights $u_n,v_n$ with sup norm at most $1$, for injective functions $f$ and $g$ on the supports of the weights, and for any bracket polynomial $\wp$ of complexity $\delta$. A direct consequence, displayed in (2.1), is the new estimate $\sum_{n_1+n_2+n_3=N}\mu(n_1n_2n_3)\ll_C N^2(\log N)^{-C}$ for the ordinary ternary Möbius sum. The paper further shows that, when $\wp$ is linear, stronger conditional bounds follow from hypotheses on the zeros of Dirichlet $L$-functions (Theorem 2.2), and that a short-interval version holds for ordinary polynomial phases with $H\ge N^{5/8+\varepsilon}$ (Theorem 2.4).
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On the paper's own terms, the discovery is that the sparse set $\{f_{k^2}:k\in\mathbb N_0\}$ behaves like a two-term additive basis for the primes in a strong density sense. The paper constructs a modulus $M$ and a residue class $n_0\pmod M$ with the property that any representation of such an $n$ forces the prime to lie in a single residue class and both square indices to lie in two prescribed residue classes modulo $192$; intersecting that class with a further modulus $769$ produces an arithmetic progression with no representations at all. Averaging $r(n)$ over the residue class gives a positive first moment asymptotic to a constant times $x$, while a sieve bound on coincidences gives a second moment of order $x$; because the second-moment constant is small enough, both $\{n:r(n)=1\}$ and $\{n:r(n)\ge2\}$ acquire positive asymptotic density within the class, hence in the integers.
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In Section II, the paper claims: 'we propose an alternative framework for extending prime-length CAZAC sequences to non-prime lengths. The proposed construction ensures that the inner product between cyclically shifted versions of the extended sequences is either zero or a constant value for any arbitrary length N.' The framework constructs length N=Q1+Q2 sequences by concatenating prime-length CAZAC blocks, and Lemma 1 shows that non-overlapping appended segments give zero inner product for distinct cyclic shifts. If correct, this gives arbitrary-length CAZAC design with controlled orthogonality.
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Theorem 1.1 states that with the explicitly displayed giant integers M and N, the number M k plus N belongs to B for every integer k at least 0, meaning no term is expressible as p plus F_m or q plus L_n. If true, this proves the existence of a single arithmetic progression of positive integers avoiding both representations simultaneously.
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Theorem 1.1 (Smooth weighted PNT and ZFR): If zeta(sigma + i t) is nonzero for sigma > 1 - eta(log|t|), then Delta(x) is bounded by x exp(-(1-epsilon) omega_eta(x)); conversely, if Delta(x) is bounded by x exp(-(1-epsilon) varpi(x)) and eta satisfies lim eta'(u) = 0, then zeta has no zeros in sigma > 1 - eta(log|t|) for large t. This equivalence is the load-bearing assertion.
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The paper's central assertion is that failure of the Goldbach conjecture is governed by the density of primes, not by their specific identities. For an even number $2n$, define $A_n$ as the set of distances $n-q$ for elements $q\in Q$ at or below $n$, and $B_n$ as the distances $q-n$ for elements $q\in Q$ between $n$ and $2n$. A representation $2n=q_1+q_2$ exists exactly when $A_n\cap B_n$ is nonempty. Taking both sets to be random subsets of $\{0,\ldots,n-1\}$ of size roughly $n/\ln n$, the paper bounds the probability of disjointness by $\exp(-n/\ln^2 n)$ and, summing over all $n\ge N$, obtains the tail bound $e^{-N^{\alpha}}$. Because the calculation depends only on the counting estimate $\pi_Q(n)\approx n/\ln n$, the paper states the result for every prime-like set $Q$, proves that shifted prime sets $P_c=\{p+c\}$ inherit Goldbach from the primes, and proposes the generalized conjecture that every such set eventually represents all sufficiently large even integers.
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The paper's central computational discovery is that the set $A=\{a: a^2+1\text{ is prime}\}$ has the Goldbach property through $a=2.5\times10^{14}$: every element of $A$ with $1<a\le 2.5\times10^{14}$ is the sum of two elements of $A$. This is established by a segmented three-stage sieve that produces the complete list of primes $m^2+1<6.25\times10^{28}$, a list occupying more than 30 terabytes. The paper reports the count $\pi_q(6.25\times10^{28})=5{,}342{,}656{,}862{,}803$, extending earlier tables, and observes that the counts track the Hardy-Littlewood/Bateman-Horn heuristic $g(x)=\frac{C_q}{2}\operatorname{li}(\sqrt{x})$ closely at the new bound. The theoretical half introduces 'Goldbach champions' $a_n$ for which the look-back statistic $j(a_n)$ sets a new record, and proves conditional results: under Hypothesis H, $j(a_n)>1$ infinitely often and $j(a_n)=1$ infinitely often; under Bateman-Horn, $\limsup j(a_n)=\infty$.
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The central claim is Theorem 1: for any finite shift set $H$ and any small prime set $P$, the limit $\kappa_H^P=\lim_{x\to\infty}\frac{1}{x}\sum_{n\le x}\prod_{h\in H}\lambda_P(n+h)$ exists and equals $\prod_{p\in P}(1-2\eta_p^H)$, where $\eta_p^H=d/(p+1)$ whenever $p$ divides none of the differences $h_i-h_j$, and exceptional primes have explicitly computable constants. The proof works through the density of the set $N_H^P=\{n:\Lambda_H^P(n)=-1\}$, which is shown to be approximable by finite unions of arithmetic progressions whose moduli use only primes from $P$. Theorem 2 then says that if some non-empty $H$ has $|\kappa_H^P|=1$, then $P$ is empty, so non-constant functions never produce perfect correlation; the proof reduces this to the two-point case via a symmetric-difference closure property. Finally, the spectrum $\Gamma_H$, the closure of all values $\kappa_H^P$ over small $P$, is exactly $[\alpha_H,1]\cup[0,1]$.
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The paper's central claim is Theorem 1.2: if some fixed $c_2>0$ has the property that $L(s,\chi)$ never vanishes in $\Re(s)>1-c_2$ for any Dirichlet character $\chi$, then for $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ and base $g$ sufficiently large, almost all even numbers $m \in [X,X+H]^*$ are Goldbach numbers; equivalently, the set of even exceptions has size $o(|[X,X+H]^*|)$. The proof follows the classical circle-method scheme for Goldbach in short intervals: the unit circle is split into Farey arcs, the major arcs give the main term $\mathfrak{S}(-2n) M^*(2n)$ built from Ramanujan sums, and the minor arcs are controlled by combining an upper bound for the weighted prime exponential sums with an $L^1$-average bound for the missing-digit exponential sum. A supporting result, Theorem 1.1, is unconditional: for $X^{3/5+\varepsilon} \ll H \ll X^{1-\varepsilon}$, the average of $d_4(n)$ over $[X,X+H]^*$ is $O((\log X)^7 |[X,X+H]^*|)$. This divisor bound is used to show that the average of a Ramanujan-sum tail in the major-arc analysis is $o(|[X,X+H]^*|)$, which is what makes the 'almost all' conclusion quantitative.
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On its own terms, the paper's central claim is Theorem 1.2: under GRH there is an effectively computable $N_0$ such that every even $N\ge N_0$ admits $1\le a,b<N$ with $a+b=N$ and $\lambda(a)=\lambda(b)=-1$. The result is derived from the stronger Proposition 1.4, which asserts that for every sufficiently large prime $N$ and every pattern $(\eta_1,\eta_2)\in\{-1,+1\}^2$ there are $\gg N e^{-C(\log\log N)^6}$ integers $n<N$ with $\lambda(n)=\eta_1$ and $\lambda(N-n)=\eta_2$. The author is careful to record that full GRH is not needed: a zero-free rectangle of the form $\operatorname{Re}(s)>1-(\log N)^{-c}$ with $c<3/50$ and $|\operatorname{Im}(s)|\le(\log N)^3$ would suffice, though with a weaker exponent.
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The paper's central discovery is Theorem 1.5: for $\varepsilon\in(0,1/3]$, $A,K\ge 1$, with $z_0=\exp(25000A^3K(\log 2A)^2/\varepsilon)$, $M$ divisible by $P(z_0)$ with all prime factors below $z_0$, and $N\ge (Mz_0)^{1+\varepsilon^{-1}}$, every subset $\mathcal{P}^*$ of primes in $[\sqrt N,N]$ with $|\mathcal{P}^*|\ge N/(K\log N)$ can be written as $1_{\mathcal{P}^*}=f^\flat/(V(z_0)\log N)+f^\sharp$. The flat function $f^\flat$ vanishes unless $(\ell,M)=1$, satisfies $0\le f^\flat(\ell)\le 2(1+\varepsilon)^2$, and reproduces the original prime sum at every rational $a/M$ up to the factor $V(z_0)\log N$; the sharp part $f^\sharp$ has Fourier transform of modulus less than $|\mathcal{P}^*|/A$ at every $\alpha$. The paper also proves cusp-counting and cusp-propagation theorems (Theorems 1.2 and 1.4) describing the set of large Fourier values before the decomposition is built. The author presents the decomposition as a general form of known local-model expansions for the primes, now valid on the full unit circle and for the original, un-W-tricked characteristic function.
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The central discovery is Theorem 3.6. If $S$ is an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group, then every $f\in S[G]$ with $|\operatorname{supp}(f)|>1$ is a sum of at most two irreducibles if and only if $\mathscr{A}_+(S)=S^\times$, and this condition is also equivalent to the weaker-looking bound at most $n$ irreducibles for any fixed $n>1$. Necessity is forced by a multi-term obstruction: a non-unit additive atom $s$ makes $sx^{g_0}+sx^{g_1}+sx^{g_2}$ impossible to write as two irreducibles. Sufficiency is constructive: using the Furstenberg splitting of each coefficient into a unit plus a remainder, the proof redistributes the unit parts into two monolithic polynomials whose top and bottom coefficients are units, which Lemma 3.1 certifies as irreducible. The same mechanism, with three pieces instead of two, yields the series-level analogue in Theorem 4.6, and Corollary 4.7 extends it to all finitely generated torsion-free groups.
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For the action of the dihedral group $D_n$ on $\mathbb{Z}_{2n}$ by $\rho^k\cdot x=x+2k$ and $\sigma\cdot x=-x$, the paper studies $S$-invariant functions: permutations $f$ for which there is an automorphism $\varphi$ of $D_n$ with $f(g\cdot x)=\varphi(g)\cdot f(x)$ for all $g,x$, where $S$ is the list of point stabilizers. The central discovery is that every such function is an affine map $x\mapsto ux+a$ with $u\in\mathrm{GL}(\mathbb{Z}_{2n})$, so the group $\widehat{\mathrm{Aut}}(D_n)_S$ is isomorphic to $\mathrm{Aff}(\mathbb{Z}_{2n})=\mathbb{Z}_{2n}\rtimes\mathrm{GL}(\mathbb{Z}_{2n})$ (Corollary 3.24). The paper reaches the same structure for both odd and even $n$. It then defines the Goldbach sieve on $\mathbb{Z}_N$ by selecting dihedral groups so that the complement of the covering is the set of residues representing Goldbach pairs, and shows that the sieve's symmetry group $G_N$ is a subgroup of $\mathrm{Aff}(\mathbb{Z}_N)$; the final sections give numerical criteria and two conjectures connecting $G_N$ to Goldbach's strong conjecture.
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By defining circles of partition as a combinatorial-geometric structure, the circle embedding method shows that every sufficiently large natural number n can be expressed as the sum of two elements from any set H with natural density strictly greater than 1/2. This is used to give asymptotic proofs of the binary Goldbach conjecture, that every large even integer is the sum of two primes, and the Lemoine conjecture.
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Stated on its own terms, the paper's central claim is the analytical aVGC: for every pair of finite positive integers $a\ge b\ge 0$ there is a finite positive integer $L(a,b)=L(b,a)$ such that for all integers $n>L(a,b)$ one can write $2n=p^{a+1}(x)+p^{b+1}(y)$ with the two summands distinct, where $p^{k}(x)$ denotes the $x$-th prime after applying the prime-indexing function $k$ times, so $0$-primes are ordinary primes and $1$-primes are prime-indexed primes. The paper reports verified thresholds $L(0,0)=3$, $L(1,0)=3$, $L(2,0)=2564$, $L(1,1)=40306$, $L(3,0)=125771$, $L(2,1)=1765126$, $L(4,0)=6204163$, $L(3,1)=32050472$, $L(2,2)=161352166$, and $L(5,0)=260535479$, obtained by one scan program up to $2n=10^{10}$. It identifies the $(1,1)$ case with a conjecture already published elsewhere, and it adds an inductive variant iVGC whose formulas $f_{x1}(a,b)$ and $f_{x2}(a,b)$ predict magnitudes, together with a step-4 heuristic $h(a,b)$ that gives nearer estimates of about $1.7\times10^9$ for $L(4,1)$ and $8.8\times10^9$ for $L(3,2)$.
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The central assertion is Theorem 1.1: if $H<X$ and $\log H/\log_2 X\to\infty$, then $\sum_{h\le H}|\sum_{p\le X}\mu(p+h)|=o(H\pi(X))$; and if $H=X^\theta$ for $\theta\in(0,1)$, the same sum is $\ll_{\theta,\delta}H\pi(X)/(\log X)^{1/3-\delta}$ for every $\delta>0$. Thus the folklore conjecture that $\sum_{p\le X}\mu(p+h)=o(\pi(X))$ holds for almost all shifts $h\le H$, with at most $O_A(H(\log X)^{-A})$ exceptions for any fixed $A$. The same machinery yields averaged Chowla-type correlations along the primes, higher correlations with products of von Mangoldt and divisor functions, and a qualitative averaged shifted-prime cancellation for arbitrary bounded multiplicative functions that do not pretend to be a Dirichlet character, when $H=X^\theta$.
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By placing every even number into one of 75 structures via the S.C.E model and applying the inequality x / ln x ≤ π(x) ≤ 1.2251 x / ln x for x ≥ 17, the strong Goldbach conjecture holds for 74 structures; the dominant structure reduces to three unproven inequalities on the model's elements.
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The central claim is that, for $s\ge 12$, the weighted prime solution count $R(P)$ for the system has the asymptotic form $R(P)\sim C P^{s-5}$ with $C>0$ whenever the system has a nontrivial real solution and, for every prime $p$, the local system has a solution with all coordinates nonzero modulo $p$. The constant is uniform over all admissible choices of the integer coefficients $u_i,v_i$. The proof isolates the positivity of the singular series as the only genuine arithmetic obstruction: the singular integral is inherited from the integer-variable case, the minor arcs contribute only $O(P^{s-5}(\log P)^{-E})$, and local solvability at every prime is turned into $S>0$. Theorem 1.2 then gives a checkable sufficient local criterion: both equations solvable modulo 2, the quadratic equation solvable modulo 3, and at least seven nonzero coefficients of each form modulo every odd prime.
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The central discovery is an asymptotic rigidity theorem for prime-generated semigroups. Writing $e_n$ for the embedding dimension and $\pi$ for the prime-counting function, the paper proves $\pi(u_n)\sim 3n$, $e_n\sim 2n$, and $u_n\sim 3p_n$. The engine is a strong three-prime theorem with almost equal summands: every sufficiently large odd $N$ is $q_1+q_2+q_3$ with $\lvert q_i - N/3\rvert\le N^\theta$ for any $\theta>11/20$. This proves Lemma 3, that every odd $N\ge(3+\varepsilon)p_n$ is in $S_{n+1}$ for large $n$, which combines with the elementary lower bound $f_n\ge 3p_n-6$ to give the Frobenius asymptotics under the parity condition, and yields the corollary $S_n=S_n^{3+\varepsilon}$ that drives the atom count. The paper then proves unconditionally that Wilf's inequality holds for every $S_n$, using the $f_n<3p_n$ criterion together with an explicit bound $f_n<n^2$ for $n\ge429$ obtained from a general Frobenius bound and standard prime estimates.
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The central claim is that the connected-sum factorization of knots mirrors the prime factorization of integers closely enough to support a bijection between knots and positive integers except 2: the trefoil is assigned 1 but is related to 2, the figure-eight knot is assigned 3, and every other knot receives a distinct integer. Under this assignment the $\star$ operation behaves like multiplication on the related numbers, while the $\times$ operation creates composite knots that fill rooms at composite positions. Theorem 8 asserts that the assignment at the $n$th induction step, the block through $2^n$, effectively determines the distribution of prime numbers in that block and is itself determined by the assignments of the previous $n-1$ blocks through preordering sequences and chains of transition. From this, Theorem 9 claims that every even number at least 6 can be written as the sum of two prime numbers, indeed as the sum of two primes each belonging to a twin-prime pair, and Theorem 10 claims that infinitely many twin-prime pairs exist.
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The paper's central claim is Theorem 20: the nontrivial zeros of the Riemann zeta function all lie on the critical line $\operatorname{Re} z = \frac12$. The claimed proof shows that the Hilbert-Polya operator for the zeros is the Virasoro energy operator with central charge $c = \frac12$, so the zeros are eigenvalues $E_j$ of that self-adjoint operator. The supporting construction is a classification table in which knots are one-to-one with integers, prime knots are bijectively assigned to prime numbers, and the trefoil plays the role of the prime 2. The table is then combined with a trace formula, patterned on the quantum-chaos trace formula, in which quantum knots act as periodic orbits and prime knots carry the amplitudes that the von Mangoldt-Selberg formula assigns to primes. The same reasoning is extended to Dirichlet $L$-functions and Dedekind zeta functions.
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For a fixed admissible tuple $H=\{h_1,\ldots,h_k\}$, the paper replaces the sharp Selberg weights by smoothed weights defined on a semigroup of short intervals. The diagonalisation $J=\sum_K \Phi(K)\Xi(K)^2$ reduces the smoothed square sum to a main term plus a controllable error, and the same structure, with one residue class removed, gives a lower bound for $\sum_{N<n\le 2N}\varpi(n+h)\gamma(n;H)\tilde\Lambda_R(n;H,\ell)^2$ with error $E_h(N;H)$. The authors' conclusion is conditional: if $E_h(N;H)\ll N/(\log N)^{C_1}$ for $R=N^{\theta/2}$ with an absolute $\theta>1/2$, uniformly in $h$, then Lemma 3 and Lemma 4 together imply $\liminf_{n\to\infty}(p_{n+1}-p_n)<\infty$. In the addendum they record that the distribution estimate needed for this implication has since been established.