---

Claim. Call a triple $1\le i<j\le n/2$ bad when no prime $p\ge i$ divides both $\binom ni$ and $\binom nj$, so that Problem 699 asks whether bad triples exist. Theorem 1.2 of the manuscript Partial Progress on Erdős Problem #699 (25 July 2026, the Overleaf project linked above; the card is Partial Progress on Erdős Problem #699) proves that no bad triple has $i=1$, $i=2$ or $i\ge1476$, and that for each fixed $i$ with $4\le i\le1475$ there are only finitely many bad triples, with no effective bound. Every possible counterexample therefore has $i=3$ or lies in a finite set that the manuscript does not determine. The argument localizes the prime powers of the rough part of $\binom ni$ by Kummer's theorem, which assigns each prime power to a point of the triangle $\Delta_i=\{(r,s):r+s<i\}$ of residue offsets, turns a polynomial vanishing to high order at every point of $\Delta_i$ into an upper bound on that rough part (an explicit degree-17 line--conic polynomial at $i=4$, a weighted line arrangement for $i\ge5$), and applies the $S$-part theorem of Bugeaud, Evertse and Győry for the fixed-index finiteness; for large $i$ an orbit polynomial with a Jacobi discriminant forces the common divisor to be large while Kummer's formula bounds it through the primes below $i$, which confines $n$, and an explicit prime in $(n-i,n]$ then divides both coefficients.

A later manuscript in the same project, Binomial coefficients sharing a large prime divisor (added to the project after the posting and announced in a comment of 2026-07-28, and dated 2026-09-21; the card is Binomial coefficients sharing a large prime divisor), works with the strict threshold $p>i$. Its Lemma 2.1 gives $G>e^{-2i}(n/i)^{i/4}$ for the common divisor $G$ when $4\le i<j\le n/2$, by a Toeplitz determinant of shifted binomial coefficients; its Theorem 3.1 puts the largest prime factor of $G$ at least $\min(q,ci\log i)$ for an absolute $c>0$ and the largest prime $q\le n$; its Corollary 4.1 leaves only finitely many strict-threshold exceptions with $i\ge121$, and its Theorem 4.2 lowers that bound to $4$: all but finitely many triples with $4\le i<j\le n/2$ have a common prime factor $p>i$. The manuscript states, without the calculation, that no counterexample has $i\ge1000$.

Submission note. Posted to erdosproblems.com as a proof claim by Wouter Van Doorn and Stefano Rocca (account ster) on 25 July 2026, giving "ChatGPT 5.6 Sol Pro" as the AI used:

For every $n$, an admissible counterexample $(n,i,j)$ can occur only when $i=3$, or in a finite set of exceptional triplets with $4\le i\le1475$ (with this finite set not explicitly determined). The argument starts from Kummer's localization of the relevant prime powers. For $i=4$ an explicit line-conic test is used, while for $i\ge5$ a weighted line arrangement gives a uniform fat-point divisor. An $S$-part theorem then yields ineffective finiteness. For the tail, an orbit polynomial and its discriminant force the common divisor of the two binomial coefficients to be large, while Kummer's formula bounds it using only primes below $i$. Comparing these bounds relegates any hypothetical counterexample, and then explicit prime-gap estimates produce a prime in $(n-i,n]$, which must divide both binomial coefficients. Wouter and I are currently working to digest, verify, and polish the proofs. Notes: We will try our best to produce a Lean verification soon, but I am afraid it will have to be conditional upon certain results like [OeSHP14]. I have deliberately tried to keep the manuscript concise in order to improve readability and avoid unnecessary verbosity at this preliminary stage. An effort has also been made to locate the various results used in the proof within the existing literature and to provide the relevant references.

Covers. No counterexample with $i=1$, $i=2$ or $i\ge1476$; for each fixed $i\ge4$ only finitely many counterexamples, without an effective bound; the case $j\le3i/2$ for every $i$ (Proposition 2.4 of the first manuscript). The case $i=3$ is not settled, and the finite exceptional sets at $4\le i\le1475$ are not determined.

Depends on. No page of this wiki.

Claimants and system. Stefano Rocca submitted the claim on 2026-07-25 for themself and Wouter van Doorn; the tab names ChatGPT 5.6 Sol Pro, and the submission said the authors were still digesting and verifying the proofs. The manuscripts state where their results come from. The first says, directly after its global theorem, that all results and arguments specific to its solution, the global theorem and the supporting lemmas included, follow Price's Overleaf project Common Prime Divisor of Binomial Coefficients (2026), its reference [Pri26]. The second declares that its proof was found by ChatGPT 5.6 Sol and then simplified and generalized by the authors, so that the end result is human-written. On 2026-07-28 van Doorn added the note that became the second manuscript, describing it as a simplified, human-written version of the ChatGPT proof with a stronger conclusion for large $i$; a commenter on 2026-07-30 observed that the exceptional range of $i$ had narrowed from $[3,1475]$ to $[3,1000]$. The submission said a Lean verification was intended, conditional on the published exhaustive prime-gap computation; none is linked.

Standing. Pending. The problem page (edited 19 July 2026, before the claim) does not credit the manuscripts, no publication exists, and both cards stand at author-recorded: the stated theorems and their cited inputs agree with their papers, but the computations behind the proofs are not retained, so the proofs count as not verified.