Erdos Problem 126: https://www.erdosproblems.com/126
been experimenting with GPT-5.5 Pro on Erdős Problem #126. I’m not claiming it solved the problem, hence the title.
What happened is that I went back and forth with the model for a while. I gave it the problem, it developed a few possible approaches, I kept asking it to push the most promising ones, and eventually it produced what looks like a candidate height-dependent partial result.
I could not find this in the existing literature either, and on the Erdos website, no partial result has been found.
I had another fresh 5.5 Pro chat verify line by line, and it said it genuinely survived its audit.
The original problem is height-free, so this would not settle it. But the last write-up it gave claims something like the following:
Let \(A\subset\mathbb N\) be finite, \(|A|=n\), and let \(S\) be the set of all primes dividing at least one off-diagonal pair-sum \(a+b\), \(a\ne b\). Let \(k=|S|\), and define the primitive height
\[
H(A)=\frac{\max A}{\gcd(A)}.
\]
The claimed partial result is that there is an absolute constant \(C\) such that
\[
n\le C(k+1)(1+\log H(A))(1+\log\log(3H(A))).
\]
It also claims a secondary statement: if \(A=\{a_1<\cdots<a_n\}\) is primitive and \(n\ge k+1\), then
\[
a_n\le a_{k+1}^{k+1}.
\]
If correct, it would say that any exponential-size counterexample has to have absurdly large primitive height, roughly double-exponential in \(k\).
I have posted links to the singular response by the model and the entire chat history, if you guys also want to check that out.
I’m posting this specifically for HUMAN proof-checking.
The last response/write-up is the main one to look at; I also posted the entire back-and-forth, since it could be useful, as it shows how the argument developed.
SINGLE chat where the candidate's partial result is written out: https://chatgpt.com/s/t_69f02c082500819188341babb6cb86e8
FULL chat history: https://chatgpt.com/share/69f025a2-9a8c-8325-99a4-4fe2254fc61c