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 $n\geq 1$ . 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)?

Source: erdosproblems.com/283

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

The site's label is PROVED (LEAN). Graham's Theorem 1 of 1963 (J. Austral. Math. Soc., refereed) settles $p(x)=x$ for every $m>77$ , with $77$ excluded; Alekseyev's Theorem 1 of 2019 (a chapter of an edited Princeton University Press volume) states $p(x)=x^2$ for every $m>8542$ (a pending claim: a book chapter with no refereeing recorded); van Doorn's unrefereed binomial-case manuscript claims the families $x+b$ ( $1\le b\le5000$ ), $5x+b$ ( $1\le b<375$ , $5\nmid b$ ) and $x^2+b$ ( $1\le b\le800$ ). The general case rests on an argument generated by the AI system GPT 5.5 Pro at the prompting of Liam Price and edited by Kevin Barreto (site thread, 3 May 2026), for the stronger form with $1$ replaced by any positive rational $\alpha$ ; the site's curator accepted it (page edited 10 May 2026, with his own proof summary in the thread), and a public Lean 4 formalization of the argument exists at the commit the formal-conjectures file pins. No refereed publication, arXiv posting or review outside the site's thread of the general argument was found. The standing in the frontmatter derives from the claim pages: the full claim Price 2026, accepted on the curator's review alone, and the partial claims Graham 1963, Alekseyev 2018 and van Doorn 2025; the argument's provenance is recorded below without judgment. The site's Lean suffix is a catalog label explained under Formalization and the Lean label.

Gr63: Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. 3 (1963), no. 4, 435--441, DOI 10.1017/S1446788700039045 (Crossref record accessed). Theorem 1 (p. 435), Theorem 3 (pp. 439--440) and the Remarks with conjecture $2'$ (p. 441). Library home: Graham 1963.
Al19: Alekseyev, M. A., On partitions into squares of distinct integers whose reciprocals sum to 1. In: The Mathematics of Various Entertaining Subjects, Volume 3 (J. Beineke and J. Rosenhouse, eds.), Princeton University Press, 2019, 213--221; arXiv:1801.05928v2 (23 April 2018, 7 pages; the chapter was not compared). Theorem 1, p. 1 of the preprint. Library home: Alekseyev 2019.
vD25: van Doorn, W., Partitions with prescribed sum of reciprocals: asymptotic bounds. arXiv:2502.02200 (v1 4 February 2025; v2 23 July 2025, 12 pages). Preprint. Theorem 1, p. 2. The site's reference gives the title with "sum of rationals". Library home: Doorn 2025.
Ca60: Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) 21 (1960), 111--124. Cited by the site and the monograph for the completeness of the values $p(n_i)$ over distinct $n_i$ without the reciprocal condition. Library home: Cassels 1960 (this page consumes no statement from it).
Gr64: Graham, R. L., Complete sequences of polynomial values. Duke Math. J. 31 (1964), 275--285. Theorem 1 is the completeness criterion that the general argument and its formalization use as their external input. Library home: Graham 1964.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 32. Library home: Erdős and Graham 1980.
vD25b: van Doorn, W., The binomial case of Graham's conjecture on polynomial representations with prescribed sum of reciprocals. Manuscript, 11 pages, a PDF in the author's GitHub repository Woett/A-normal-paper-is-probably-fine, linked from the thread on 31 August 2025 as work in progress; the file of 24 March 2026 is the one the claim page pins. Theorems 1--2 (the reduction), pp. 2--7; Theorems 3--5 (the families), pp. 7--8. Not refereed; not on arXiv; not filed in the library.
Manuscript: "Polynomial Egyptian Sums: a formalization-informed revised presentation", dated 6 May 2026, 13 pages, the PDF the thread's comment of 6 May 2026 links on Google Drive (file id 1cW2Z7vpTjLQ2Wf6SMb6_nbO9JYlfznnt), accessed 2026-09-18. Its title page names the AI system as author; Theorem 8 (p. 4) is the main theorem. Not a refereed source; not filed in the library. The original writeup's Overleaf read link (gdmnffbshxsq) shows no PDF without a login (2026-09-18); the Drive file is the copy this page cites.
OEIS: van Doorn, W., Sequence A380791, The On-Line Encyclopedia of Integer Sequences (2025): the number of positive rationals $x$ whose threshold $k(x)$ equals $n$ ; accessed.

Problem 283

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