GPT 6 Astra Ultra attempt on Math-6191

Erdős Problem 1137 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Write $p_n$ for the $n$th prime and $d_n=p_{n+1}-p_n$. Determine whether \[ \frac{\max_{2\leq n<x}d_nd_{n-1}}{\left(\max_{1\leq n<x}d_n\right)^2}\longrightarrow0 \qquad(x\to\infty). \]

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

The displayed lower index bounds make the adjacent gaps defined under one-based prime indexing; finite initial-index conventions do not affect the limiting question.

ATTACK PLAN AND WORK

Put $D_N=\max_{1\le j\le N}d_j$ and introduce \[ A_N=D_N^{-2}\max_{2\le j\le N}d_{j-1}d_j, \qquad B_N=D_N^{-1}\max_{2\le j\le N}\min(d_{j-1},d_j). \] Then the following exact comparison holds: \[ B_N^2\le A_N\le B_N. \tag{1}\] Indeed, for positive $u,v\le D_N$, $\min(u,v)^2\le uv\le D_N\min(u,v)$; taking maxima gives $1$. Consequently the original limit is zero if and only if $B_N\to0$. Equivalently, for every $\epsilon>0$, all sufficiently long prime prefixes have no adjacent gaps both at least $\epsilon D_N$.

This isolates the required local exclusion. A bound on the frequency of large gaps alone cannot give it: even one exceptional adjacent pair affects the maximum. Nor does divergence of $D_N$ suffice for arbitrary sequences. For the positive sequence $g_{2j-1}=g_{2j}=j$, the corresponding ratio is exactly one at every even prefix. This example is not a model of prime gaps or a counterexample to the problem; it identifies precisely what an argument using only maxima would miss.

An exact finite check can compute $D_N$, the largest adjacent product, and the largest adjacent minimum simultaneously with one pass through a prime list. The accompanying local check verifies $1$ on sieved prime prefixes; no finite range establishes the limit.

FINITE COMPUTATIONAL CHECK

For successive prime-value cutoffs $10^3,10^4,10^5,10^6$, the numbers of complete gaps are $167,1228,9591,78497$; the maximum gaps are $20,36,72,114$. The adjacent-product ratios are respectively $9/25,32/81,203/432,50/171$. Their variation is not evidence of monotonicity or a proof of a limit.

VERIFICATION AND REMAINING GAP

Prove the adjacent record-scale exclusion for prime gaps. Known or computed large individual gaps do not supply this exclusion. The next useful input must control two neighboring gaps relative to the largest gap in the entire prefix.

SOURCES

https://www.erdosproblems.com/1137

https://raw.githubusercontent.com/google-deepmind/formal-conjectures/main/FormalConjectures/ErdosProblems/1137.lean

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$8\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.