Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306
Abstract
We give an elementary proof that every positive rational number a/b with b squarefree is a finite sum of distinct unit fractions 1/n, where each n is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in (y2,2y2], together with $2$ and the primes of b, and a tuned initial segment of the primes in (y8,y9], and show that some subgraph has reciprocal sum congruent to a/b modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.
Related
- An elementary treatment of Ramanujan's elementary continued fractions
- Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
- An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
- An elementary proof of the 2-regularity of Pythagorean triples
- An elementary proof of Brémont's theorem on the convergence of equilibrium measures at temperature zero
BibTeX
@misc{li2026unit,
title = {{Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306}},
author = {Shisheng Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32140},
eprint = {2609.32140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32140},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306
AU - Shisheng Li
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32140
UR - https://arxiv.org/abs/2609.32140
AB - We give an elementary proof that every positive rational number a/b with b squarefree is a finite sum of distinct unit fractions 1/n, where each n is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in (y^2,2y^2], together with $2$ and the primes of b, and a tuned initial segment of the primes in (y^8,y^9], and show that some subgraph has reciprocal sum congruent to a/b modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.
ER -