Erdős Problem #283 — Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$.

The conjecture asks whether every sufficiently large integer can be represented by polynomial values at distinct positive integers whose reciprocals sum to $1$. It is recorded as Erdős Problem $\#283$; the retrieved sources do not give a posing date.

Known results

• Graham proved the linear case $p(x)=x$, in a stronger form for every positive rational reciprocal sum and lower bound.
• Cassels proved that the polynomial hypotheses suffice without the reciprocal-sum constraint.
• Burr proved a power-polynomial variant when repeated indices are allowed.
• Alekseyev proved $p(x)=x^2$ for every $m>8542$; van Doorn proved many linear and quadratic cases, including $p(x)=x+5$ and $p(x)=x^2+100$.

Undated GPT-5.5 Pro proof claim

A discussion reports that GPT-5.5 Pro, prompted by Price, produced a proof of the stronger statement with $1$ replaced by any rational $\alpha>0$. This remains an AI-generated claim: no retrieved preprint or independently verified proof corroborates it.

Current status (as of March 2026): Special cases are established, but the full conjecture has no verified proof; the reported GPT-5.5 Pro proof remains unconfirmed.