Larsen 2026 sufficiently abundant numbers pseudoperfect

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corollary_5: States that there is an absolute constant C such that every positive integer n with sigma(n) at least Cn is a sum of distinct proper divisors of itself, deduced from Theorem 1 of the same paper.

theorem_1: States that for every positive epsilon there is an integer L such that every integer n with sigma(n)/n greater than 2 + epsilon and no prime factor less than L is a sum of distinct proper divisors of itself.

theorem_4: States the paper's general Egyptian-fraction theorem: for a set D of products of an element of B with divisors of a product of pairwise coprime integers in dyadic blocks, minus a set E, and a target l/k with w(D)(l/k)^-1 in a prescribed window, at least two subsets of D have reciprocal sum congruent to l/k modulo 1, provided Hypothesis 3 holds.

theorem_6: States that for every partition of the perfect squares greater than one into two non-empty parts there are non-empty finite subsets of the two parts with the same reciprocal sum, the affirmative answer to the squares case of Problem 318.

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Daniel Larsen, Sufficiently abundant numbers are pseudoperfect. Manuscript, 9 pages, posted in the author's GitHub repository Larsen-Daniel/Erdos-318 as 318.pdf. The repository was created on 31 January 2026; the file was uploaded that day with the commit message "This is most of a proof" and replaced on 1 February 2026 (commit 39139e2b, the repository's head on 2026-09-18). The paper carries no date of its own; its references cite the site's problem pages as accessed and the PDF's creation stamp is 1 February 2026, so the year in the slug is 2026. It is not on arXiv (API searches) and has no journal record (Crossref bibliographic query of 2026-09-18); it is unrefereed.

The copy read for this card is that 9-page file, obtained from https://github.com/Larsen-Daniel/Erdos-318/blob/main/318.pdf, the link the site's Problem 318 thread gives (comment of 1 February 2026); 291,983 bytes; a fresh fetch of the raw file at commit 39139e2b on 2026-09-18 returned identical bytes. An earlier 8-page version of the same title (PDF creation stamp 30 January 2026), whose abstract states only the pseudoperfect result and which has no Theorem 6, was read for comparison. The text layer is clean; the statements below were read in it, and Theorems 1 and 6 and the closing acknowledgment on the page images of pp. 1, 8 and 9. No notice is printed in the manuscript (pp. 1--2 and 8--9 read); the hosting repository (https://github.com/Larsen-Daniel/Erdos-318, read 2026-10-02) holds only the PDF and its TeX source, has no LICENSE file, and its About panel reads "No description, website, or topics provided.", so no terms are stated; the term is unstated.

Read status: claims checked. Theorem 1 (p. 1), the definition of $w$ (p. 2), Hypothesis 3 and Theorem 4 (p. 5), Corollary 5 (p. 7) and Theorem 6 (p. 8) were read clause by clause; the proof of Theorem 6 (pp. 8--9) was read for structure; no proof was checked.

Provenance the paper declares: its closing line (p. 9) acknowledges the assistance of two named AI systems "for proofreading". This card records the declaration and does not evaluate it.

Contents

Notation (pp. 1--2): a number is pseudoperfect when it is a sum of distinct proper divisors of itself (Sierpiński); $\sigma(n)/n$ is the abundance index; $w(A)=\sum_{1<a\in A}1/a$; $\mathrm{Div}(N)$ is the set of divisors of $N$ and $\mathrm{Div}^*(N)$ the divisors greater than $1$; $x\sim y$ means $x\in[y,2y)$. The introduction attributes the general strategy (a weighted random selection of divisors analyzed by the circle method) to a suggestion of Tao and Bloom, with the work of Croot, of Bloom and of Conlon et al. as examples, and the Egyptian-fraction recasting to Friedman.

• Theorem 1 (p. 1): for every $\varepsilon>0$ there is an integer $L$ such that every integer $n$ with $\sigma(n)/n>2+\varepsilon$ and no prime factor below $L$ is pseudoperfect. Proof pp. 2--7: dyadic blocks of prime factors, a greedy approach to $1$ with slack, a pull-back (Lemma 2), then Theorem 4; result page theorem_1.
• Hypothesis 3 and Theorem 4 (stated on p. 5, proof pp. 5--7): the circle-method theorem. For a set $D=B\cdot\mathrm{Div}(\prod_{q\in Q}q)\setminus E$ built from blocks $Q_i\subseteq[y_i,2y_i]$ of prescribed sizes, whose union $Q$ consists of pairwise coprime integers, and a set $B$ of divisors of $(y_1^2)!$, and for $\ell/k\in(0,1]$ with $\alpha=w(D)(\ell/k)^{-1}$ in a window $[1+\epsilon/100,\log^{1+\epsilon}y_1]$, there are at least two subsets $D'\subseteq D$ with $w(D')\equiv\ell/k\pmod1$, provided also that $k$ divides the least common multiple of $D$, that $B$ contains $1$ and its elements are coprime to every element of $Q$, that $E\subseteq B$ contains the elements of $B$ below $y_1^2$, that the $y_i$ meet the theorem's size and spacing conditions, and that Hypothesis 3 holds for $y=y_1$; result page theorem_4, which also states Hypothesis 3.
• Corollary 5 (p. 7): there is an absolute constant $C$ such that every positive integer $n$ with $\sigma(n)\ge Cn$ is a sum of distinct proper divisors. This answers the Benkoski--Erdős question, the site's Problem 825 (row below); result page corollary_5. The proof fixes $\varepsilon=1/10$ in Theorem 1, takes the $L$-rough part $m$ of $n$ and notes that if $m$ is not pseudoperfect then $\sigma(n)/n\le(\sigma(m)/m)\prod_{p<L}p/(p-1)\ll1$.
• Theorem 6 (p. 8): whenever the perfect squares above $1$ are split into two non-empty classes $X$ and $Y$, some non-empty finite $X'\subseteq X$ and $Y'\subseteq Y$ have $w(X')=w(Y')$; result page theorem_6. The paragraph after it says the theorem answers a question of Erdős and Graham, that Sattler claimed a positive resolution in the 1980s but never wrote down the argument, that Graham had shown by combinatorial methods that every rational in the expected intervals is a sum of reciprocals of finitely many distinct squares, and that Meza suggested Bloom's methods might apply.

Compiled scope

Statements only, as listed; the proofs of Theorems 1 and 6 were not checked and nothing here is independently reviewed. The paper is an unrefereed manuscript with a declared AI-assistance acknowledgment. The site's Problem 318 page accepts its squares result ("Larsen has proved that the answer is yes in the case of squares excluding $1$"; page last edited 1 April 2026).

Bears on. #318: Theorem 6 is the problem's third question, the squares excluding $1$, restated for the partition $X=f^{-1}(1)$, $Y=f^{-1}(-1)$ of a non-constant sign function $f$; the theorem page writes out the equivalence. #825: Corollary 5 (p. 7, read on the page image) states an absolute $C$ such that every positive integer $n$ with $\sigma(n)\ge Cn$ is a sum of distinct proper divisors, which contains the problem's statement with $\sigma(n)>Cn$; it is deduced from Theorem 1 in four lines (page corollary_5; Theorem 1 on theorem_1); the statement was checked, the proof of Theorem 1 was not, and the manuscript is unrefereed. Theorem 4 (page theorem_4) is the circle-method step of the proofs of both Theorem 1 and Theorem 6 and states neither problem itself.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.