Square products of factorials and a conjecture of Erdős and Graham
Abstract
For n≥2 let F(n) be the least k≥2 such that n! is the largest factor in a product of k distinct factorials that is a perfect square, and let Dk(X) be the number of n≤ X with F(n)=k. Erdos and Graham asked for the order of growth of Dk(X) for 3≤ k≤6, and conjectured that D6(X)≫ X. We prove that D3(X)=κ3√X+Oε(X2/5+ε) with an explicit constant κ3=2.7097…, and that D5(X)asymp D6(X)asymp X. Together with classical facts, this determines the order of growth of Dk(X) for every k. The exponent 2/5 comes from balancing a uniform bound for Pell equations against Gallagher's larger sieve, with residue restrictions supplied by the Weil bound. For five and six factors we restrict to integers with a prime factor exceeding X1−α, where α>0 is small and fixed. We exclude shorter representations by combining an equidistribution estimate for primes of Matomaki, Radziwill, Shao, Tao and Teravainen with the large sieve. In an appendix we use a zero-sum theorem for finite abelian groups to construct, for every m≥2, perfect m-th powers that are products of a bounded number of distinct factorials with arguments given by fixed affine functions.
Related
- The Burr-Erdős-Graham-Sós conjecture for the seven-cycle
- Structural rigidity in the Erdős-Graham two-set permutation problem
- A counterexample to the Chung-Graham-Spiro gap-set conjecture
- Graham's rearrangement conjecture in finite abelian groups
- Local Search for Almost-Spanning Square Grids in Erdős--Rényi Random Graphs
BibTeX
@misc{yudin2026square,
title = {{Square products of factorials and a conjecture of Erdős and Graham}},
author = {Fedir Yudin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01899},
eprint = {2610.01899},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01899},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Square products of factorials and a conjecture of Erdős and Graham
AU - Fedir Yudin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01899
UR - https://arxiv.org/abs/2610.01899
AB - For n≥2 let F(n) be the least k≥2 such that n! is the largest factor in a product of k distinct factorials that is a perfect square, and let D_k(X) be the number of n≤ X with F(n)=k. Erdos and Graham asked for the order of growth of D_k(X) for 3≤ k≤6, and conjectured that D_6(X)≫ X. We prove that D_3(X)=κ_3√X+O_ε(X^2/5+ε) with an explicit constant κ_3=2.7097…, and that D_5(X)asymp D_6(X)asymp X. Together with classical facts, this determines the order of growth of D_k(X) for every k. The exponent 2/5 comes from balancing a uniform bound for Pell equations against Gallagher's larger sieve, with residue restrictions supplied by the Weil bound. For five and six factors we restrict to integers with a prime factor exceeding X^1−α, where α>0 is small and fixed. We exclude shorter representations by combining an equidistribution estimate for primes of Matomaki, Radziwill, Shao, Tao and Teravainen with the large sieve. In an appendix we use a zero-sum theorem for finite abelian groups to construct, for every m≥2, perfect m-th powers that are products of a bounded number of distinct factorials with arguments given by fixed affine functions.
ER -