GPT 6 Astra Ultra attempt on Math-5526

Erdős Problem #242: independent continuation

Date: 23 September 2026. Source versions read in full: 242.tex in gpt_pro_5.4.

1) OBJECTIVE AND SOURCE AUDIT

The target is $4/n=1/x+1/y+1/z$ with $x<y<z$ positive integers, for every $n>2$. The supplied congruence identities are correct; the surviving $n\equiv1\pmod{24}$ condition does not itself generate denominators. I replace further ad hoc congruence guessing by a complete divisor test for each fixed $n$, and apply it to survivors of the supplied tests.

2) WORK: AN EXACT DIVISOR CRITERION

There is a solution if and only if there is an integer \[ n/4<x<3n/4 \] and positive divisors $d<e$ of $N^2$, where $N=nx$ and $m=4x-n$, such that \[ de=N^2,\qquad d\equiv e\equiv-N\pmod m,\qquad (d+N)/m>x. \] The corresponding denominators are $y=(d+N)/m$, $z=(e+N)/m$. Indeed, after removing $1/x$ the equation becomes $m/N=1/y+1/z$, which is equivalent to \[ (my-N)(mz-N)=N^2. \] Each factor is positive because $m/N>1/y$ and $m/N>1/z$. The congruences make $y,z$ integral, $d<e$ makes them distinct, and the last inequality enforces the order. Conversely the factor identity expands to exactly the required unit-fraction equation. Finally, $4/n>1/x$ and $4/n<3/x$ give the strict range for $x$.

Exact arithmetic gives \[ \begin{array}{c|ccc} n&x&y&z\\\hline 73&20&210&30660\\ 97&25&810&392850\\ 193&50&1380&1331700 \end{array} \] For example, for $n=73,x=20$, $m=7,N=1460$ and the factors are $d=10$, $e=213160$. All three displayed rows were independently checked as rational identities. Each $n$ is $1\pmod{24}$, and respectively $n+1=2\cdot37,2\cdot7^2,2\cdot97$; the middle example is already handled by the source's divisor criterion, while the first and third are not.

3) ADVERSARIAL VERIFICATION

The factor test is finite for a fixed $n$ but its search range grows with $n$. Exhibiting solutions for these values proves no uniform coverage of the surviving residue class. The factorization identity is standard algebra, not a claimed new resolution of the Erdős--Straus conjecture. The companion exact checks are in verification/root_checks.py.

7) FINAL AND REFERENCE

UNRESOLVED. I obtain a complete finite criterion and explicit certificates beyond some of the supplied modular cases. The missing step is a proof that a suitable divisor pair exists for every surviving $n$. The distinct-denominator convention was checked in the indexed official statement, https://www.erdosproblems.com/242.

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.