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Goldbach's Conjecture
Formal claims (Lean)
Stated claims
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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 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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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 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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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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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.