gpt 6 astra pro attempt on Math-171

Cramer Conjecture on the Limsup of Normalized Prime Gaps

Definitions and mathematical statement

Let $p_n$ be the $n$th prime and $g_n=p_{n+1}-p_n$. With natural logarithms, the selected assertion is the exact constant statement \[ \limsup_{n\to\infty}\frac{g_n}{(\log p_n)^2}=1. \] Equivalently, for every $\varepsilon>0$, all sufficiently large $n$ satisfy $g_n\le(1+\varepsilon)(\log p_n)^2$, while infinitely many satisfy $g_n\ge(1-\varepsilon)(\log p_n)^2$. The equality of the limsup is materially stronger than an unspecified upper bound $g_n=O((\log p_n)^2)$. This section records the catalogue's precise constant-one version; the title alone should not be used to interchange different prime-gap conjectures or heuristics.

Formulation and concept references: [S1].

Short English statement

Is the largest recurring scale of consecutive prime gaps exactly the square of the logarithm of the smaller prime, with limiting upper constant one?

Research attempt

Current frontier.

The constant $1$ is the decisive issue. Granville's small-prime-conditioned model instead suggests recurring gaps with normalized size at least $2e^{-\gamma}>1$; this is a heuristic, not a theorem about the actual primes [E1], pp. 23--24. Established landmarks include the Baker--Harman--Pintz prime-in-short-interval estimate and the Ford--Green--Konyagin--Maynard--Tao large-gap lower bound [E2], [E3]. Neither reaches the required logarithmic-square scale with a positive limiting constant. A currently accessible manuscript entitled Improved long gaps between primes claims a stronger iterated-logarithm lower bound [E4]; it is not independently verified here, and even its stated bound is $o((\log X)^2)$. It must not be reported as a resolution of the exact constant-one target.

Attack route.

Three routes are natural: construct covering congruences giving longer prime-free intervals; estimate correlations of many prime translates; or transfer the extreme-value calculation from a random model. The first must produce interval length comparable to the square of the logarithm of the constructed location. The third requires a genuine arithmetic transfer theorem, not just numerical agreement. I pursue the second, retaining the alternating cancellations that are normally lost in a termwise error bound. The question becomes how accurately high factorial moments determine an extremely rare empty window.

Attempt: exact finite identities.

Let $X\ge2$ and $h\ge1$ be integers, and put \[ \mathcal I_X=\{X,X+1,\ldots,3X\},\qquad N_X=2X+1, \qquad A_m=\pi(m+h)-\pi(m). \] Define \begin{align} E(X,h)&=\#\{m\in\mathcal I_X:A_m=0\},\label{wsref-0-33-30-30-30-36-37-36-31-3a-65-6d-70-74-79}\\ F_j(X,h)&=\sum_{m\in\mathcal I_X}\binom{A_m}{j} =\sum_{1\le a_1<\cdots<a_j\le h} \#\{m\in\mathcal I_X:m+a_1,\ldots,m+a_j\text{ prime}\}. \label{wsref-0-33-30-30-30-36-37-36-31-3a-66-61-63-74-6f-72-69-61-6c} \end{align} Here $F_0=N_X$. The second equality simply counts pairs consisting of a starting point and $j$ prime locations. It is exact, including the inadmissible patterns whose contribution is zero.

Set $B_M=\sum_{j=0}^M(-1)^jF_j$. For each nonnegative integer $a$, \[ \sum_{j=0}^M(-1)^j\binom aj= \begin{cases} 1,&a=0,\\ (-1)^M\binom{a-1}{M},&a\ge1, \end{cases} \] with binomial coefficients zero beyond their usual range. This follows from Pascal's identity by telescoping. Therefore \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-31-3a-62-6f-6e-66-65-72-72-6f-6e-69} B_{2r+1}\le E(X,h)\le B_{2r}. \end{equation} Thus a short list of fixed-order correlations is not the relevant object: the truncation order will have to grow with $X$.

There is also an exact relation to consecutive prime gaps: \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-31-3a-67-61-70-2d-69-64-65-6e-74-69-74-79} E(X,h)=\sum_{p<q\ {\rm consecutive}} \max\{0,\min(3X,q-h-1)-\max(X,p)+1\}. \end{equation} Indeed, $(m,m+h]$ contains no prime precisely when the preceding prime $p\le m$ and the next prime $q$ satisfy $m+h<q$. Each admissible $m$ is counted exactly once.

Attempt: a quantitative rare-window criterion.

For fixed $c>0$ take $h=\lfloor c(\log X)^2\rfloor$ and use the varying local intensity \[ \lambda_m=\int_m^{m+h}\frac{\,\mathrm d t}{\log t},\qquad Q(X,h)=\sum_{m\in\mathcal I_X}e^{-\lambda_m}. \] Uniformly for these $m$, $\lambda_m=c\log X+O_c(1)$, whence \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-31-3a-70-6f-69-73-73-6f-6e-2d-73-63-61-6c-65} Q(X,h)\asymp_c X^{1-c}. \end{equation} The integral is a comparison model, not an asserted prime-counting formula. It avoids the spurious first-moment discrepancy introduced by freezing $\log m$ at $\log X$ across an entire interval of length $2X$.

Write \begin{align} P_M&=\sum_{m\in\mathcal I_X}\sum_{j=0}^M \frac{(-\lambda_m)^j}{j!},\\ \mathcal D_M&=B_M-P_M =\sum_{j=0}^M(-1)^j \left(F_j-\frac1{j!}\sum_m\lambda_m^j\right). \label{wsref-0-33-30-30-30-36-37-36-31-3a-73-69-67-6e-65-64-2d-64-69-73-63-72-65-70-61-6e-63-79} \end{align} Taylor's theorem gives the rigorous error bound \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-31-3a-74-61-79-6c-6f-72} |P_M-Q(X,h)|\le N_X\frac{\lambda_{\max}^{M+1}}{(M+1)!}. \end{equation} Choose a constant $A=A(c)$ so large that $A\log(A/(ec))>3$, and take $M=A\log X+O(1)$ with either prescribed parity. Using $n!\ge(n/e)^n$ and $\lambda_{\max}=c\log X+O_c(1)$, the right side of (30006761:taylor) is $o(1)$.

The following two arithmetic estimates would imply the printed target: \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-31-3a-6d-69-73-73-69-6e-67} \begin{aligned} c>1:&\quad \mathcal D_M=o((\log X)^2) &&\text{for the even choice of }M,\\ 0<c<1:&\quad \mathcal D_M=o(Q(X,h)) &&\text{for the odd choice of }M. \end{aligned} \end{equation} These are unproved hypotheses, not consequences of the preceding identities. Notice that they control the signed discrepancy itself, not the sum of absolute errors of its terms.

Here is the full implication, including the uniform upper bound. For $c>1$, (30006761:bonferroni)--(30006761:missing) give $E(X,h)=o((\log X)^2)$. If a prime $p\in[X,2X]$ had next-prime gap $g>h+\delta(\log X)^2+2$, then the first $\lfloor\delta(\log X)^2\rfloor$ starts after $p$ would be empty windows. For large $X$ all lie in $[X,3X]$, contradicting that estimate. Given $\varepsilon>0$, take $c=1+\varepsilon/3$ and $\delta=\varepsilon/3$. Applying the conclusion on successive dyadic intervals proves $g\le(1+\varepsilon)(\log p)^2$ for every sufficiently large prime $p$.

For $0<c<1$, the odd bound gives $E(X,h)\ge(1+o(1))Q(X,h)>0$. Choose an empty window and its preceding prime $p\le3X$. The next-prime gap is greater than $h$, and hence its normalized size is at least $h/(\log(3X))^2=c+o(1)$. These preceding primes tend to infinity as $X\to\infty$. Letting $c$ approach $1$ from below proves the required lower limsup. This proves the conditional implication, not (30006761:missing).

Stress test: where the proposed transfer breaks.

Replacing (30006761:signed-discrepancy) by the triangle inequality is catastrophic. Terms near $j\approx\lambda_m$ have size roughly $e^{\lambda_m}$ before multiplication by $N_X$, whereas their alternating sum has size $e^{-\lambda_m}$. Relative error estimates for individual terms need not survive this cancellation. Nor does a Poisson limit for fixed $h/\log X$ apply when that ratio is $c\log X\to\infty$.

Small-prime obstructions are material: two adjacent integers above $2$ cannot both be prime, whereas the independent model assigns a positive pair probability. Averaging admissible singular series can correct some fixed-order statistics, but no growing-order cancellation theorem with (30006761:missing)'s accuracy has been supplied. Granville's calculation is specifically a warning that the constant-one transfer may be false [E1]. Proving the first line of (30006761:missing) in ignorance of that warning would effectively assume the desired upper extreme-value law.

The direct counterexample route also fails at its scale. The elementary composites $N!+2,\ldots,N!+N$ certify only a length of order $N$ at a location of logarithm of order $N\log N$. That certified lower bound, divided by the squared logarithm, tends to zero; this statement is about the certificate, not an upper bound for the actual surrounding prime gap. The accompanying script checks (30006761:bonferroni) and (30006761:gap-identity) at $X=100,h=24$. These are finite consistency checks, not evidence establishing either limit in (30006761:missing).

Outcome.

Outcome: Exploratory approach with unresolved gap.

The exact empty-window identities and the implication from (30006761:missing) are proved. The attempt isolates the needed cancellation scale and shows why fixed-order moments, mean density, and naive factorial constructions do not reach it. Neither the constant-one upper bound nor infinitely many actual prime gaps exceeding that constant has been established. The competing arithmetic heuristic remains a reason to investigate disproof rather than to presume the catalogue's equality.

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Sources

• S1 Weisstein, E. W., Cramer Conjecture, From MathWorld - A Wolfram Resource, updated 16 August 2026. https://mathworld.wolfram.com/CramerConjecture.html. Catalogue formulation source.
• E1 Andrew Granville, "Harald Cramér and the distribution of prime numbers," Scandinavian Actuarial Journal 1995, no. 1, 12--28, especially pp. 23--24. https://dms.umontreal.ca/~andrew/PDF/cramer.pdf. Small-prime-conditioned heuristic; not a proof of a prime-gap limsup.
• E2 R. C. Baker, G. Harman and J. Pintz, "The difference between consecutive primes, II," Proc. London Math. Soc. $3$ 83 $2001$, 532--562. Main prime-in-short-interval result. https://doi.org/10.1112/plms/83.3.532. Established polynomial-scale upper-bound landmark, not a logarithmic-square estimate.
• E3 Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, "Long gaps between primes," J. Amer. Math. Soc. 31 $2018$, 65--105; arXiv:1412.5029. Theorem 1. https://arxiv.org/abs/1412.5029. Established large-gap lower bound.
• E4 OpenAI, "Improved long gaps between primes," eight-page online manuscript, introductory theorem; accessed 27 September 2026. The consulted PDF does not supply a journal publication record. https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps.pdf. Current proof claim used only to compare the stated scale with the target; not independently verified or endorsed.