Problem 283

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claims/: The 4 claim pages of Problem 283, one per claimant's result; the problem's standing derives from them.

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Statement. 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

$$m=p(n_1)+\cdots+p(n_k)?$$

Formulation. The site's wording as of 2026-09-18 (page last edited 10 May 2026). The polynomial takes integer values at the integers and its leading coefficient is positive; "no $d\ge2$ with $d\mid p(n)$ for all $n\ge1$" says that the values at positive integers have no common divisor other than $1$, the condition the 1980 monograph writes as $\gcd(p(1),p(2),\ldots)=1$ and Graham's 1963 conjecture $2'$ writes prime by prime. The denominators are distinct positive integers whose reciprocals sum to exactly $1$, and the number $k$ of them may depend on $m$. The question is whether every sufficiently large integer $m$ is the sum of $p$ over the denominators of such a representation. The cases $p(x)=x$ (Graham 1963, refereed) and $p(x)=x^2$ (Alekseyev 2019, a published book chapter) are recorded below (Graham's accepted, Alekseyev's pending); the general case is the question.

Status. 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.

Source. erdosproblems.com/283, accessed 2026-09-18: the problem page (PROVED (LEAN), whose label tooltip reports an affirmative solution with a proof verified in Lean; source key [ErGr80, p. 32]; last edited 10 May 2026; a formalized statement recorded; OEIS A380791 linked), its ten-comment discussion thread (11 August 2025 to 10 May 2026) and its empty proof-claim tab. The site cites [Gr63], [Ca60], [Al19] and [vD25] in its commentary and thanks Wouter van Doorn and Liam Price. Cite as: T. F. Bloom, Erdős Problem #283, https://www.erdosproblems.com/283, accessed 2026-09-18.

References.

• [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_theorem_partitions.
• [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_partitions_into_squares_distinct_integers_whose.
• [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_partitions_prescribed_sum_reciprocals_asymptotic_bounds.
• [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_representation_integers_as_sums_distinct_summands (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_complete_sequences_polynomial_values.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [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.

Formalization. Statement only, with a pointer to an external proof. The file ErdosProblems/283.lean of formal-conjectures at the linked revision (main) declares erdos_283 : answer(True) ↔ ∀ p : ℚ[X], Condition p under category research solved with proof sorry, where Condition p says: if $p$ takes integer values at all integers, has positive leading coefficient and no $d\ge2$ divides $p(n)$ for all $n\ge1$, then for all sufficiently large integers $m$ there are $k\ge1$ and $0=n_0<n_1<\cdots<n_k$ with $\sum_{i=1}^k1/n_i=1$ and $\sum_{i=1}^kp(n_i)=m$. Its formal_proof attribute points to lines 9738--9746 of Erdos/P283/Proof_flat.lean in the repository Shashi456/erdos-formalizations at the commit the Price claim page's formalization link pins. Seven variants, all sorry, record Graham's case, the rational-$\alpha$ form, Cassels's completeness statement, Burr's power case with repetitions, Alekseyev's threshold and van Doorn's families $p(x)=x+b$ ($1\le b\le5000$) and $p(x)=x^2+b$ ($1\le b\le800$). The community database records formal_status Lean, as of its last update of 10 May 2026, the statement as formalized, as of its last update of 6 November 2025, OEIS A380791 and no formal-proof URL. This corpus has built and audited none of these files; see Formalization and the Lean label below.

Current assessment

The question (site formulation of 2026-09-18). The statement above; PROVED (LEAN); last edited 10 May 2026; source key [ErGr80, p. 32]. The commentary, in summary: Graham [Gr63] settled the case $p(x)=x$ and asked whether the same holds when the reciprocal sum $1$ is any positive rational $\alpha$ (the threshold for $m$ then depending on $\alpha$); Cassels [Ca60] showed that the two hypotheses on $p$ already make every large integer a sum of values $p(n_i)$ at distinct $n_i$, the reciprocal condition dropped; Burr handled the powers $p(x)=x^k$ when repeated denominators are permitted; Alekseyev [Al19] settled $p(x)=x^2$ from $m=8543$ on, illustrated by $1=\frac12+\frac14+\frac16+\frac1{12}$ with $200=2^2+4^2+6^2+12^2$; van Doorn [vD25] studied the size of the threshold for $p(x)=x$ and, per the thread, settled a large number of linear and quadratic polynomials, $p(x)=x+5$ and $p(x)=x^2+100$ among them; and the general case, strengthened to every positive rational $\alpha$, is credited to a proof given by GPT 5.5 Pro at Price's prompting, with the thread holding a summary. The thread (ten comments): van Doorn's summary of Graham's paper and of his own bounds (11 August 2025), his reduction of the binomial case $p(x)=ax^d+b$ to a finite search with the linear and quadratic families above (31 August 2025, a manuscript in progress on GitHub), his asymptotic for $n_{\alpha,m}$ with a conditional formalization (28 March 2026), Price's announcement (3 May 2026), Barreto's note on his edits (3 May), Nat Sothanaphan's summary of the argument (3 May) and his report of 6 May that he had confirmed it, the formalizer's comments of 6 May with the Lean file, the later formalization of its one axiom and an updated PDF, and Thomas Bloom's proof summary (10 May). The thread also links chat-transcript pages, which are not sources. The community database lists the problem as proved (Lean), as of its last update of 10 May 2026.

Origin. Printed p. 32 of the 1980 monograph, with $\mathscr X$ the set of finite sets $\{x_1,\ldots,x_n\}$, $0<x_1<\cdots<x_n$, with $\sum1/x_k=1$. The authors recall Graham's theorem [Gr (63) b] that every $m\ge78$ is $\sum x_k$ over some set in $\mathscr X$, and that $m=77$ is not, and then pose the conjecture: "It seems highly likely that for any polynomial $p:\mathbf Z\to\mathbf Z$ it is true that for all sufficiently large $m$, there is a set $\{x_1,\ldots,x_t\}\in\mathscr X$ with $\sum_{k=1}^tp(x_k)=m$, provided $p$ satisfies the obvious necessary conditions: (i) the leading coefficient of $p$ is positive; (ii) $\gcd(p(1),p(2),\ldots)=1$." They add that by Cassels [Cas (60)] the two conditions suffice for every large integer to be a sum $\sum p(a_i)$ over distinct $a_i$, and that Burr [Burr $(\infty)$] showed that for every $k$ every large integer is a sum $\sum x_i^k$ over a tuple in $\mathscr X'$, the version with repetitions allowed. The conjecture is the case $\alpha=\beta=1$ of Graham's conjecture $2'$ of 1963 (Remarks, p. 441): condition 2 of his Theorem 3 could be replaced by $n=f(a_1)+\cdots+f(a_k)$ for any polynomial $f$ mapping integers to integers with positive leading coefficient such that for every prime $p$ some $f(m)$ is not divisible by $p$; Graham adds that very little was then known about the problem.

Partial results. Each addresses instances of the problem's quantifier over $p$ and has its own partial claim page; Graham's is accepted, Alekseyev's and van Doorn's are pending. Graham's Theorem 1 (p. 435; claims checked; J. Austral. Math. Soc., refereed; claim page Graham 1963): every integer $n>77$ is $a_1+\cdots+a_k$ with $1<a_1<\cdots<a_k$ and $\sum1/a_i=1$, the case $p(x)=x$; the Remarks record Lehmer's unpublished check that $77$ has no such partition. His Theorem 3 (pp. 439--440) is the rational-$\alpha$ form of the same case: for positive rationals $\alpha,\beta$ every large $n$ is a sum of distinct integers exceeding $\beta$ with reciprocal sum $\alpha$. Alekseyev's Theorem 1 (preprint p. 1; claims checked; a chapter of an edited volume, with no record that it was refereed; claim page Alekseyev 2018): $8542$ is the largest integer that is not a sum of squares of distinct positive integers whose reciprocals sum to $1$, the case $p(x)=x^2$; the paper's computation was not rerun. Van Doorn's Theorem 1 (preprint p. 2; claims checked) quantifies the case $p(x)=x$: with $n_\alpha$ the least integer beyond which every integer has a partition into distinct parts with reciprocal sum $\alpha=p/q$, $n_\alpha<cq\log^3q/(\min(\alpha,1)\log\log q)+c(e+\epsilon)^{2\alpha}$, and his Theorem 2 gives $n_\alpha>0.3q\log^2q$ for some $\alpha$ with any large prime denominator $q$ in any interval; $n_1=78$. That paper is a preprint and settles no instance of the question. Cassels's theorem (completeness of the values $p(a_i)$ without the reciprocal condition) is recorded as the site's and the monograph's attribution, and this page consumes no statement from his paper; Burr's result is cited by the monograph as unpublished. Van Doorn's binomial-case manuscript ([vD25b]; claim page van Doorn 2025; claims checked, proofs not verified) reduces, for $f(x)=ax^d+b$ with coprime coefficients and any $\alpha>0$, the existence of the threshold $n_0(f,\alpha)$ to a finite computation (its Theorems 1 and 2) and carries that computation out for $\alpha=1$ in its Theorems 3--5: $n_0(x+b,1)\le172+10b$ for $1\le b\le5000$, $n_0(5x+b,1)\le20000$ for $1\le b<375$ with $5\nmid b$, and $n_0(x^2+b,1)\le50000$ for $1\le b\le800$, the source of the site's examples $p(x)=x+5$ and $p(x)=x^2+100$; a manuscript in progress, not on arXiv.

The general case: the site-accepted AI-generated argument (provenance recorded, not judged). The claim page Price 2026 records this result. On 3 May 2026 Liam Price wrote in the thread that GPT 5.5 Pro, prompted by him, had resolved the problem for every positive rational $\alpha$, linking the Overleaf writeup, and that Kevin Barreto had helped clean up the proof and had noticed that Problem 351 follows; Barreto wrote that he kept the changes minimal. Bloom's summary of 10 May 2026, in the thread: fix $p(x)=cx^d+\cdots$, put $u_j=36j+1$ and $D_j=u_ju_{j+1}$, and define $q(j)$ by $p(6D_j)+p(3D_j)+p(2D_j)-p(D_j)=q(j)$, a polynomial of degree $2d$ in $j$ with leading coefficient $c(6^d+3^d+2^d-1)\,36^{2d}$; by Theorem 1 of Graham [Gr64] there are $g,X$ such that every multiple of $g$ that is at least $X$ is a sum of distinct values $q(j)$; by telescoping, $\frac1{36}=\sum_{0\le j<J}\frac1{D_j}+\frac1{36u_J}$; choose finite sets $A_k$ with $\sum_{a\in A_k}1/a=1-\frac1{36}$, avoiding the classes $\{0,1,2,3,6\}$ modulo $36$, with $\sum_{a\in A_k}p(a)\equiv k\pmod g$ (a finite computation for fixed $p$); then $N_J^k=\sum_{0\le j<J}p(D_j)+p(36u_J)+\sum_{a\in A_k}p(a)$ is representable, $N_{J+1}^k-N_J^k=c\,36^{2d}J^{2d}+O(J^{2d-1})$, which is smaller than $q(J)$ for large $J$, and $|N_J^k-N_J^{k'}|\ll1$; switching a denominator $D_j$ to $2D_j,3D_j,6D_j$ keeps the reciprocal sum $1$ and the distinctness and changes the $p$-sum by $q(j)$; so for a large $N$ in the class $k$ modulo $g$ there is a $J$ with $X<N-N_J^k<q(J)$, and writing $N-N_J^k=\sum_{j\in S}q(j)$ by Graham's theorem forces $S\subseteq\{0,\ldots,J-1\}$, so that switching the $D_j$, $j\in S$, reaches $N$. (The thread post prints the leading coefficient of $q$ as $c(6^d+3^d+2^d-1)$, the base term as $p(u_J)$ and the step as $cJ^{2d}$, and chooses $J$ with $2cJ^{2d}\ge N-N_J^k>X$; the forms above carry the factor $36^{2d}$ and the term $p(36u_J)$ that the definitions give.) Bloom describes the local switches through $1=\frac12+\frac13+\frac16$ as the natural idea and finds the writeup more complicated than it needs to be. Sothanaphan's summary of 3 May describes the same structure (the Roth--Szekeres--Graham result applied to $A(x):=p(2x)+p(3x)+p(6x)-p(x)$) and says he had not examined the technical steps; on 6 May he wrote that he had confirmed the proof, through a transcript link, which is not a source. The manuscript at the thread's Drive link ("Polynomial Egyptian Sums: a formalization-informed revised presentation", 6 May 2026, 13 pages) states as Theorem 8 (p. 4): for $\alpha\in\mathbb Q_{>0}$, $L\ge1$ and $p\in\mathbb Q[x]$ integer-valued on $\mathbb Z$ with positive leading coefficient and no fixed divisor on the positive integers, there is $m_0$ such that every integer $m\ge m_0$ is $\sum_{i=1}^kp(n_i)$ with distinct $L<n_1<\cdots<n_k$ and $\sum1/n_i=\alpha$; its Theorem 1, the Roth--Szekeres--Graham completeness theorem (Graham's 1964 Theorem 1), is "the single external input"; its Appendix A lists twenty-eight presentation changes made for the formalization and says the proof strategy is unchanged; the formalizer's comment says nothing changed from the original proof. The argument is consumed at the level of these summaries and the theorem statement; no step is verified. Acceptance evidence: the site's label and commentary, and the community database's status. No refereed publication, arXiv posting or review outside the site's thread of the general case was found (search scope below). The corpus records this as the site's acceptance of an AI-generated argument with a public formalization, and does not judge the argument.

Formalization and the Lean label. The site's Lean suffix is a catalog label. Behind it, as of 2026-09-18: (1) the formal-conjectures statement above, whose formal_proof attribute pins a commit of Shashi456/erdos-formalizations. (2) The file Erdos/P283/Proof_flat.lean at that commit (dated 14 May 2026; the file itself last changed on 7 May 2026): 11,229 lines, one import Mathlib, no sorry and no axiom declaration; its header calls it a standalone flat bundle for Problems 283 and 351, states the trust boundary as Mathlib's core axioms propext, Classical.choice, Quot.sound, to be verified "with #print axioms at the bottom" (the file at this commit contains no such command), and attributes the proof of 3 May 2026 to the AI system with the human cleanup. Lines 9738--9746 declare theorem theorem_1 (α : ℚ) (hα : 0 < α) (L : ℕ) (hL : 1 ≤ L) (p : ℚ[X]) (hp : IntValued p) (h_lead_pos : 0 < p.leadingCoeff) (h_no_fixed_div : NoFixedDivisor p hp) : ∃ m₀ : ℕ, ∀ m : ℕ, m₀ ≤ m → ∃ (k : ℕ) (n : Fin (k + 1) → ℕ), StrictMono n ∧ (L < n 0) ∧ (α = ∑ i, (1 : ℚ) / (n i)) ∧ ((m : ℚ) = ∑ i, p.eval ((n i : ℕ) : ℚ)), where IntValued p says $p$ takes integer values at integers and NoFixedDivisor p hp says no $d\ge2$ divides every value at positive integers. The file proves its roth_szekeres_graham wrapper from its own Erdos.P283.RSG.graham_complete_polynomial_values (the formalizer's comment of 6 May 2026 says the input was first an axiom and was then formalized from Graham's 1964 paper, so that the proof rests on the three classical axioms alone; a reported claim) and ends with a wrapper for Problem 351. The formal-conjectures Condition p is the case $\alpha=1$, $L=1$ of theorem_1 in form; that specialization and the fidelity of theorem_1 to the site's statement are not audited. The formalizer's comments of 6 May 2026 report that the file type-checks in Lean 4.27 and 4.28 with current Mathlib and passed an independent checker; reported, not reproduced. (3) Van Doorn's ExplicitGraham.lean in the repository Woett/Lean-files, at the linked revision of 27 March 2026: 3,879 lines, no sorry, two declared axioms (Croot_lemma, standing for Propositions 1 and 2 of Croot's 2001 paper, and smoothinarithgeneral, that smooth integers have positive density in every residue class), and the final theorem explicit_graham: for every positive rational $\alpha$, $n_{\alpha,m}/((e^{2\alpha}-1)m^2)\to1/2$. This is the threshold asymptotic for $p(x)=x$ (the thread's comment of 28 March 2026), not this problem's statement; the file also proves, in its Part 2, the lemma ogGraham, the existence part of Graham's Theorem 1, the case $p(x)=x$, recorded on Graham's claim page. (4) Two further Lean files are recorded as formalization links on the claim pages: van Doorn's ErdosProblem283.lean in the same repository, generated by Aristotle, which formalizes the two reduction theorems of his binomial-case manuscript and is re-hosted in Boris Alexeev's repository plby/lean-proofs as Erdos283b.lean, marked partial; and Alexeev's Erdos283.lean, which declares itself a formalization of the Price argument. All of these are records of the files' declarations: nothing was built or kernel-checked by this corpus and no local credit is claimed. The community database records formal_status Lean, as of its last update of 10 May 2026, and no formal-proof URL.

Forum items (leads with provenance, not status). Van Doorn's manuscript on the binomial case ([vD25b], linked 31 August 2025; the source of the site's examples $p(x)=x+5$ and $p(x)=x^2+100$), recorded under Partial results and on its claim page; his companion paper "Partitions with prescribed sum of reciprocals: computational results" (arXiv:2502.01409, cited by OEIS A380791 and by the asymptotic paper as its [11]; not held); OEIS A380791 (W. van Doorn, February 2025): the number of positive rationals $x$ whose threshold $k(x)$ equals $n$, beginning $2,2,2,1,2,4,5,5,7,7,5,12,\ldots$, with the comment that Graham proved every $k\ge78$ has a partition into distinct parts with reciprocal sum $1$; its terms are not verified.

Search scope. The site's problem, discussion and proof-claim pages; the community database record; the formal-conjectures file at the pinned commit; the GitHub API for the commit dates of Shashi456/erdos-formalizations and Woett/Lean-files and the raw Lean files at the pinned commits; the thread's Drive and Overleaf links; the arXiv abstract pages of 1801.05928 (v1 18 January 2018, v2 23 April 2018; journal reference the 2019 chapter) and 2502.02200 (v1 4 February 2025, v2 23 July 2025; no journal reference); the Crossref records for Graham's 1963 paper and Alekseyev's chapter; the Semantic Scholar citation lists of 1801.05928 (two records: van Doorn's computational paper and a 2021 paper on Graham partitions) and 2502.02200 (none); the arXiv API queries abs:partition AND abs:reciprocals AND abs:distinct (19 records; the relevant ones are Alekseyev's paper and van Doorn's two papers) and abs:polynomial AND abs:"unit fractions" AND abs:"sufficiently large" (none); OEIS A380791; the primary sources [Gr63], [Al19], [vD25] and [ErGr80] at the cited passages. Not searched: MathSciNet, zbMATH, Google Scholar, X, the general web. Nothing found adds a refereed source for the general case or disputes the accepted argument.

Remaining gaps. (1) The general case rests on an AI-generated argument accepted by the site with a public, locally unbuilt Lean formalization; a refereed publication, a review outside the site's thread or a local build would strengthen it. (2) Graham's and Alekseyev's theorems are compiled at statement level; Alekseyev's computation was not rerun and the published chapter was not compared. (3) Cassels's paper has a library card but is not consumed, Burr's result is unpublished, van Doorn's binomial manuscript is consumed at statement level and his computational paper is not held. (4) The fidelity of the pinned Lean theorem to the site's statement is not audited.

Progress and known results

• Graham (1963): Theorem 1, the case $p(x)=x$ for every $m>77$, with $77$ excluded (Lehmer, unpublished); Theorem 3, the rational-$\alpha$ form of that case; the conjecture $2'$, whose case $\alpha=\beta=1$ is this problem.
• Alekseyev (2019): Theorem 1, the case $p(x)=x^2$ for every $m>8542$, sharp.
• Cassels (1960) and Burr, as attributed by the site and the monograph: completeness of the values $p(a_i)$ over distinct $a_i$ without the reciprocal condition; the case $p(x)=x^k$ with repeated denominators.
• Van Doorn (2025, preprint): Theorem 1, $n_\alpha<cq\log^3q/(\min(\alpha,1)\log\log q)+c(e+\epsilon)^{2\alpha}$ for $\alpha=p/q$, with $n_\alpha>0.3q\log^2q$ infinitely often (Theorem 2) and $n_{\alpha,m}=(\frac12+o_\alpha(1))(e^{2\alpha}-1)m^2$ for fixed $\alpha$ (Theorem 3).
• Van Doorn (manuscript, 2025): the binomial families $x+b$ ($1\le b\le5000$), $5x+b$ ($1\le b<375$, $5\nmid b$) and $x^2+b$ ($1\le b\le800$), Theorems 3--5 of [vD25b], on the claim page van Doorn 2025.
• The general case, for every positive rational $\alpha$: the AI-generated argument of May 2026 accepted by the site, with its Lean formalization, recorded above with provenance and on the claim page Price 2026; the same argument settles Problem 351 in its corrected form (thread and manuscript).

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• cassels_1960_representation_integers_as_sums_distinct_summands
• cassels_1960_representation_integers_as_sums_distinct_summands / theorem_i
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• alekseyev_2019_partitions_into_squares_distinct_integers_whose
• alekseyev_2019_partitions_into_squares_distinct_integers_whose / lemma_2
• alekseyev_2019_partitions_into_squares_distinct_integers_whose / theorem_1
• alekseyev_2019_partitions_into_squares_distinct_integers_whose / theorem_6
• alekseyev_2019_partitions_into_squares_distinct_integers_whose / theorem_7
• doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds
• doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds / theorem_1
• doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds / theorem_2
• doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds / theorem_3
• doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds / theorem_4
• graham_1963_theorem_partitions
• graham_1963_theorem_partitions / theorem_1
• graham_1963_theorem_partitions / theorem_2
• graham_1963_theorem_partitions / theorem_3