Erdős Problem #283

declared status 'proved'. Formalized: yes. 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 $n\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\leq n_1<\cdots <n_k$ such that$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$and$$m=p(n_1)+\cdots+p(n_k)?$$ Current best: Burr has proved this if $p(x)=x^k$ with $k\geq 1$ and if we allow $n_i=n_j$. van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds. Prize: no. OEIS: A380791. Tags: number theory, unit fractions.

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