gpt 6 astra ultra attempt on Math-65

Goldbach's Conjecture

Definitions and mathematical statement

With $\mathcal P$ denoting the positive primes, the statement is \[ \forall n\in2{\mathbb{Z}},\quad n>2\quad\Longrightarrow\quad \exists p,q\in\mathcal P\text{ with }n=p+q. \] The primes are not required to be distinct, so $4=2+2$ is allowed. Every even integer is quantified, not merely all sufficiently large integers or a density-one subset.

Formulation and concept references: [S1].

Short English statement

Can every even integer larger than two be written as the sum of two primes?

Research attempt

Scope and source check.

The target includes every even integer greater than two, allows equal primes, and therefore includes $4=2+2$. An asymptotic argument for large integers needs an explicit finite-range completion before it proves this formulation. Oliveira e Silva--Herzog--Pardi [E1] report verification through $4\cdot10^{18}$; that published computation is not rerun here. It is a finite theorem, not a replacement for control of the unbounded tail.

The recent catalogue source [S2], Bhowmik--Grimmelt's August 2026 revision, is largely a survey of exceptional-set methods, with additional explicit formula work. It does not assert that the exceptional set is empty. Li--Liu [S3], version 2, state a representation $N=p+rq$, with $p,q$ prime, $r=1$ or prime, and $r\le q^{0.9}$ for sufficiently large even $N$. This still permits $r>1$, so its stated conclusion is weaker than binary Goldbach. The manuscript statements were inspected; their full proofs were not independently audited. Neither claim is used as a premise of the elementary arguments below.

The selected route asks exactly what the circle method would have to prove for one prescribed even integer, and compares that with what an average-square error bound actually guarantees. The goal is to identify the first missing uniform estimate, not to infer pointwise positivity from an exceptional set of density zero.

An exact Fourier representation without prime powers.

For an integer $X\ge4$, put $e(t)=\exp(2\pi i t)$ and define \[ S_X(\alpha)=\sum_{p\le X}(\log p)e(p\alpha),\qquad R(N)=\sum_{\substack{p+q=N\\p,q\text{ prime}}}\log p\log q. \] The sum counts ordered pairs. For $4\le N\le X$, orthogonality gives \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-66-6f-75-72-69-65-72} R(N)=\int_0^1 S_X(\alpha)^2e(-N\alpha)\,d\alpha. \end{equation} This follows by expanding a finite double sum and using $\int_0^1e(m\alpha)\,d\alpha=1$ for $m=0$, zero otherwise. All weights are positive, so $R(N)>0$ is exactly the desired existence statement. Proper prime powers are absent by definition, and no subtraction estimate for them is hidden in this equivalence.

The square in (ra:25:fourier) is not an absolute square. Replacing it by $|S_X|^2$ counts prime differences $p-q=N$. For example with $X=N=4$, the difference count is zero, whereas $R(4)=(\log2)^2>0$. Positivity of $|S_X|^2$ therefore cannot be used to prove positivity of the required Fourier coefficient. The actual coefficient is real, but substantial complex cancellation occurs inside the integral.

Local factors give a positive predicted main term.

For an odd prime $r$, count residue pairs $(a,b)$ with $a+b=N$ modulo $r$ and both coordinates nonzero. The count is $r-1$ when $r\mid N$ and $r-2$ otherwise: choose $a$, excluding zero and, in the second case, the distinct residue $N$. Relative to independent unit densities, the local multiplier is \[ \frac{r\,\#\{a\bmod r:a\ne0,\ N-a\ne0\}}{(r-1)^2}. \] For even $N$, the factor at two is two. Consequently the formal singular series can be written \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-73-65-72-69-65-73} \mathfrak S_G(N)=2C_2 \prod_{\substack{r\mid N\\r>2}}\frac{r-1}{r-2}, \qquad C_2=\prod_{r>2}\left(1-\frac1{(r-1)^2}\right). \end{equation} The infinite product is strictly positive: the factors lie in $(0,1)$, and the sum of their deficits converges by comparison with $\sum_{n\ge2}n^{-2}$. For instance $-\log(1-u)\le2u$ for $0\le u\le1/2$ bounds the logarithmic tail. Thus $\mathfrak S_G(N)\ge c_0:=2C_2>0$ for every even $N$.

This derivation is an exact local calculation, not a proof that $R(N)\sim\mathfrak S_G(N)N$. CRT similarly multiplies the residue counts over a fixed squarefree modulus, but actual primality is a global restriction. The positive lower bound for the proposed main term is useful only if an accompanying analytic error can be controlled.

The small prime two requires a separate boundary check. If $N>4$ is even and one summand were two, the other would be the even integer $N-2>2$ and could not be prime. Thus every representation for even $N>4$ uses two odd primes, consistent with the local unit calculation at two. The exceptional representation of four is handled by the exact Fourier identity and the finite check, not by an asymptotic local model. Similarly, ordered counting never doubles a diagonal pair: when $N/2$ is prime it contributes one term of weight $(\log(N/2))^2$. These conventions ensure that the positive coefficient being targeted matches the precise existence statement at both the boundary and large values of $N$.

A precise pointwise sufficient estimate.

Choose $Q=(\log X)^B$ with fixed $B>0$ and define major arcs as the union, modulo one, of \[ |\alpha-a/q|\le Q/(qX),\qquad 1\le q\le Q,\quad 0\le a<q,\quad(a,q)=1. \] Let this union be $\mathfrak M_X$, with complement $\mathfrak m_X$, and set \[ A_X(N)=\int_{\mathfrak M_X}S_X(\alpha)^2e(-N\alpha)\,d\alpha, \quad I_X(N)=\int_{\mathfrak m_X}S_X(\alpha)^2e(-N\alpha)\,d\alpha. \] The sets are invariant under $\alpha\mapsto-\alpha$, making both integrals real. They sum to $R(N)$ exactly. Their definition is valid even when arcs overlap; for large $X$ the usual disjointness also follows from $X>2Q^3$ and separation of reduced fractions.

Suppose, for some explicit $X_0$ and all $X\ge X_0$, one could establish uniformly for even $X/2\le N\le X$ that \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-74-61-72-67-65-74-73} |A_X(N)-\mathfrak S_G(N)N|\le c_0X/8, \qquad |I_X(N)|<c_0X/8. \end{equation} Then $R(N)>c_0X/4>0$. Dyadic intervals would cover every sufficiently large even integer. An explicit check of the remaining interval $4\le N<X_0$ would finish the exact catalogue statement. This implication is proved by the displayed estimates; the estimates themselves are not proved here. Classical major-arc analysis and the role of exceptional sets are discussed in [S2]. In this notebook no theorem with different uniformity or an implicit exceptional set is substituted for (ra:25:targets).

What a mean-square estimate really provides.

The elementary Bessel inequality already quantifies the loss when one passes from a single integer to most integers. Fix $X$ and suppose a major-arc lower bound $A_X(N)\ge aX$ is available on its even half-interval, for a fixed $a>0$. If such an $N$ has no representation, then $I_X(N)=-A_X(N)$, so $|I_X(N)|\ge aX$. Let $U=\sup_{\alpha\in\mathfrak m_X}|S_X(\alpha)|$. Orthogonality and Bessel give \begin{align*} \sum_{N\in\mathbb Z}|I_X(N)|^2 &=\int_{\mathfrak m_X}|S_X(\alpha)|^4\,d\alpha\\ &\le U^2\int_0^1|S_X(\alpha)|^2\,d\alpha =U^2\sum_{p\le X}(\log p)^2 \le U^2 X\log^2 X. \end{align*} The first equality is Parseval for the square-integrable function $1_{\mathfrak m_X}S_X^2$; restricting the sum to even $N$ would give an inequality, which is all we need. Therefore the number $E_X$ of exceptional even integers in that interval satisfies \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-65-78-63-65-70-74-69-6f-6e-61-6c} E_X\le\frac{U^2\log^2X}{a^2X}. \end{equation} This derivation uses only the elementary bound on the number of primes; sharper familiar second-moment estimates are unnecessary for the point.

For example, even a bound $U\le X/(\log X)^A$ with fixed $A>1$ would give only $E_X\ll_a X/(\log X)^{2A-2}$. Its density tends to zero, but the bound grows and does not force the integer $E_X$ to vanish. Applying Cauchy--Schwarz directly to one coefficient instead gives $|I_X(N)|\le U\sqrt X\log X$, which is useful for positivity only with a far stronger supremum bound. Neither estimate captures the specific oscillatory cancellation of $e(-N\alpha)$ needed pointwise.

A sharp stress test of the average-to-pointwise step.

Consider an auxiliary array with predicted main value $aX$ at every even index in $[X/2,X]$. Let its actual value equal that prediction except at one index, where it is zero. All values remain nonnegative, and the sum of squared errors is exactly $a^2X^2=o(X^3)$. Thus even a mean-square error small relative to the aggregate natural scale $X^3$ is compatible with a missing representation. Choosing one such exceptional index in each dyadic interval makes infinitely many exceptions of density zero. These arrays are not prime counts and do not disprove Goldbach; they refute the logical inference from the stated norm bound to universal positivity.

An average theorem supplemented by finitely many direct checks has the same issue: the hypothetical sparse exceptional indices may all lie beyond the checked range. To use integrality, the final upper bound on $E_X$ must actually be strictly below one for every remaining interval, or a separate argument must remove each possible exceptional integer. Calling a positive real upper bound "small" cannot replace this step.

Exact finite checks and counting conventions.

The local script builds the prime indicator by an integer sieve and computes its ordinary polynomial convolution. If $g(N)$ is the number of unordered pairs with $p\le q$, the ordered count satisfies \[ r(N)=2g(N)-1_{N/2\text{ prime}}. \] It checks this identity and verifies every even $4\le N\le10000$ has a representation. Selected values are \[ \begin{array}{r|r|r} N&r(N)&g(N)\\\hline 4&1&1\\10&3&2\\100&12&6\\1000&56&28\\10000&254&127 \end{array} \] The diagonal correction is essential at four and ten. The same script enumerates all residues modulo $210$ and checks the product of the local counts $r-1$ or $r-2$ in (ra:25:series); for example there are twenty admissible ordered residue pairs for $N=100$ modulo $210$. These exact local and finite computations do not estimate either arc integral. Reproduction files are checks.py and results.json in _Local/top_problems/continuation_15_25_2026_09_27/analytic/.

Remaining gap and outcome.

The work proves the exact prime-weighted Fourier identity, the positive local-series bound, and the exceptional-set inequality with all normalizations visible. The first unresolved analytic step is a bound for the individual minor-arc coefficient strong enough to preserve positivity, paired with a major-arc estimate of the stated uniformity. Even proving eventual positivity would leave an effective threshold and the finite interval below it to address the full target.

Outcome: UNRESOLVED. No pointwise estimate of the required strength and no counterexample even integer were obtained. The explicit norm obstruction explains why an almost-all conclusion cannot be promoted to the requested theorem. The process used direct derivation, primary source inspection, and exact finite checks; it does not claim that the open problem has been solved or independently certified.

Sources

• S1 C. K. Caldwell, "Prime Conjectures and Open Questions," The PrimePages, https://t5k.org/notes/conjectures/. https://t5k.org/notes/conjectures/. Formulation source.
• S2 The exceptional set of the Goldbach problem. https://arxiv.org/html/2607.27282v2. Further reference; not a proof endorsement.
• S3 Theorem $1+1.9$ on the Goldbach Conjecture. https://arxiv.org/abs/2606.05224. Further reference; not a proof endorsement.
• S4 Restricted Goldbach Sums in Arithmetic Progressions: Local Obstructions, an Explicit Almost-All Framework, and a General-Modulus Extension. https://www.preprints.org/manuscript/202607.2077. Further reference; not a proof endorsement.

• E1 Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot10^{18}$, Mathematics of Computation 83 $2014$, no. 288, 2033--2060, DOI: 10.1090/S0025-5718-2013-02787-1. https://www.ams.org/mcom/2014-83-288/S0025-5718-2013-02787-1/S0025-5718-2013-02787-1.pdf. Primary computational reference; that large computation was not rerun in this notebook. Consulted 27 September 2026.
• E2 Gautami Bhowmik and Lasse Grimmelt, The exceptional set of the Goldbach problem, arXiv:2607.27282v2 (13 August 2026), especially Sections 1 and 4--7. https://arxiv.org/html/2607.27282v2. Version-specific identification of [S2]: survey and new explicit-formula work, not an assertion that every even integer is represented. Consulted 27 September 2026.
• E3 Jiamin Li and Jianya Liu, Theorem $(1+1.9)$ on the Goldbach Conjecture, arXiv:2606.05224v2 (8 August 2026). https://arxiv.org/abs/2606.05224v2. Version-specific identification of [S3]: stated representation permits a semiprime summand. Statement inspected; complete proof not independently audited or used here. Consulted 27 September 2026.