Mihnea 2025 further verification empirical evidence erdos straus

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section_2: Reports a modular-filter computation extending Salez's verification of the Erdős–Straus conjecture from primes up to 10^17 to primes up to 10^18.

section_3: Reports the number of representations of 4/p as three unit fractions for the 66737 primes up to 3.5·10^7 in Mordell's six open residue classes modulo 840, split into Type-1 and Type-2 solutions.

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Spiridon Mihnea, Dumitru C. Bogdan, Further verification and empirical evidence for the Erdős-Straus conjecture. arXiv:2509.00128 (2025).

The paper extends Salez's modular-filter computation for the Erdős-Straus conjecture, that 4/n is a sum of three unit fractions, from p <= 10^17 to p <= 10^18. Adding the filter S_29 to Salez's construction yields a residue set R_8 of 2101514 classes modulo G_8 = 25878772920, roughly twice as efficient as the previous R_7, and the surviving integers are checked in batches B_k = {r + kG_8 : r in R_8} against a precomputed set of 140000 prime filters; verification to 10^18 amounts to 38641709 batches and took about two weeks using GMP arbitrary-precision arithmetic. Section 3 turns to the solution-counting function f(p) = #{(x,y,z) : 4/p = 1/x + 1/y + 1/z}, using Bradford's finite search space ceil(p/4) <= x <= ceil(p/2) with an explicit construction of y and z from a divisor d of x^2, split into Type-1 (p does not divide y) and Type-2 (p divides y) solutions, and evaluates f(p) for the first 66737 primes p congruent to 1, 121, 169, 289, 361, 529 mod 840 (the residues left open by Mordell), the primes up to 3.5·10^7 in those classes, finding 12763383 Type-1 and 5838200 Type-2 solutions in all. The conjecture is equivalent to f(p) > 0 for all p, so both parts give empirical evidence toward Problem 242 without proving it. Code is released at github.com/esc-paper/erdos-straus.

Source: https://arxiv.org/abs/2509.00128.

The copy read for this card is arXiv:2509.00128v1 (29 August 2025, 4 pages; the only version listed on 2026-09-18, with no journal reference); its title page prints the second author as Bogdan C. Dumitru. The paper is a computation report without theorem labels: its result is the authors' statement that the modular-filter run for all primes p <= 10^18 completed, and nothing here reruns it. Its convention allows repeated denominators (x, y, z positive integers). Read status: claims checked. The statements of Sections 1--3 were read clause by clause on the page images (Section 2 runs from p. 1 to p. 2, Section 3 from p. 2 to p. 3); the code was not fetched. Result pages: section_2 (the verification to 10^18) and section_3 (the solution counts). The verification is recorded for Problem 242 as a pending partial claim, Mihnea and Dumitru 2025. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2509.00128), every other right reserved.

Bears on.

• #242: the authors report that the conjecture holds for every prime p <= 10^18, so for every n with 2 < n <= 10^18 through a prime factor (Section 2); the solution counts of Section 3 are numerical data for primes up to 3.5·10^7 and settle no further n. Neither part proves the conjecture.

Results to transcribe.

• Verification bound (Section 2): The authors report that the Erdős-Straus conjecture holds for all primes p <= 10^18, extending Salez's 10^17 by adding the modular filter S_29 to obtain R_8 with 2101514 classes mod G_8 = 25878772920.
• Solution counting (Section 3): Empirical evaluation of f(p), the number of representations of 4/p as three unit fractions, for the 66737 primes up to 3.5·10^7 in the residue classes 1, 121, 169, 289, 361, 529 mod 840, split into Type-1 and Type-2 solutions via Bradford's construction.

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