GPT 6 Astra Ultra attempt on Math-751

Erdős Problem #302: independent continuation

Date: 23 September 2026. Source versions read in full: 302.tex in gpt_pro_5.4.

1) OBJECTIVE AND SOURCE AUDIT

Let $f(N)$ maximize the size of a subset of $[1,N]$ containing no three distinct denominators with $1/a=1/b+1/c$. The source supplies density $5/8$ and optimizes a single-residue low block plus a high interval. I extend its obstruction to several residue classes simultaneously and show why this broader version still cannot beat $5/8$.

2) WORK: A MULTI-RESIDUE BARRIER

Let $R$ be a set of units modulo $m\ge2$ such that \[ \gcd(r-s,m)>1\quad\hbox{for all }r,s\in R. \tag{1} \] Take \[ A=\{n\le N/4:n\bmod m\in R\}\ \cup\ [\lceil N/2\rceil,N]. \] This is solution-free. In any proposed solution the smallest denominator is $a=bc/(b+c)<N/2$, so it lies in the low block. If $b$ also lies there, the equation becomes $(b-a)c=ab$. Since $a,b$ are units modulo $m$, this congruence has a solution in $c$ only if $\gcd(b-a,m)=1$, contrary to $1$. The same argument excludes $c$ from the low block. If both $b,c$ are high, then $1/a=1/b+1/c<4/N$, hence $a>N/4$, another contradiction. Consequently \[ |A|=\left(\frac12+\frac{|R|}{4m}\right)N+O_m(1). \]

But $R$ and $R+1$ are disjoint in $\mathbb Z/m\mathbb Z$: an intersection would give two members differing by one, whose difference is coprime to $m$. Thus $2|R|\le m$, and this entire compatible multi-residue template has density at most $5/8$. The parity choice $m=2,R=\{1\}$ attains it.

3) ADVERSARIAL VERIFICATION

Condition $1$ is sufficient to forbid a low--low pair for every third residue. It is not necessary for every possible interval construction. Accordingly the upper bound just proved is a bound on this template, not on $f(N)$. It does not rule out interdependent choices of several scale intervals or more selective choices of the third denominator. No assertion that the optimal asymptotic constant equals $5/8$ is made.

7) FINAL AND SOURCE

UNRESOLVED. A larger class than the source's one-residue family has been analyzed rigorously, but the global extremal question remains. The target and the existing $5/8$ construction are taken from the supplied 302.tex; the external search returned official indexing but did not supply an additional theorem used in this proof.

8) COMPLETION ESTIMATE

COMPLETION: 10% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.