For n≥ 2, define F(n) to be the least integer r≥ 2 for which there exist integers 1≤ a_1<…<a_r-1<n such that n!,a_1!… a_r-1! is a perfect square. For r≥ 2 and X≥ 1, let D_r(X)=#n≤ X:F(n)=r. Determine the order of growth of D_r(X) as X→∞ for 3≤ r≤ 6. Erdős and Graham conjectured, in particular, that D_6(X)≫ X; the three-factor case was expected to have square-root order, D_3(X)asymp√ X.

Erdős–Graham conjecture on square products of factorials

For $n\ge 2$ , define $F(n)$ to be the least integer $r\ge 2$ for which there exist integers $1\le a_1<\cdots<a_{r-1}<n$ such that $n!\,a_1!\cdots a_{r-1}!$ is a perfect square. For $r\ge 2$ and $X\ge 1$ , let $D_r(X)=\#\{n\le X:F(n)=r\}$ . Determine the order of growth of $D_r(X)$ as $X\to\infty$ for $3\le r\le 6$ . Erdős and Graham conjectured, in particular, that $D_6(X)\gg X$ ; the three-factor case was expected to have square-root order, $D_3(X)\asymp\sqrt X$ .

References

Primary source

arXiv

Additional references

Square products of factorials and a conjecture of Erdős and Graham — arXiv — Fedir Yudin

analytic-number-theory number-theory

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

An unrefereed preprint claims the remaining three-, five-, and six-factor growth rates have been determined, but the claims have not been independently checked.

Erdős and Graham posed questions about when products of several factorials are squares and how frequently this occurs, including the three-factor case and analogous cases with more factors. Their 1976 work left the relevant growth questions open.

Known results

Erdős and Graham, 1976: conjectured a square-root-order count for the three-factor case and recorded examples, but did not prove the conjectured asymptotic.

October 2026 development

Fedir Yudin’s preprint claims the five- and six-factor growth rates and the precise asymptotic scale for the three-factor case, thereby covering the cases posed by Erdős and Graham. A related preprint gives a three-factor count of $x^{1/2+o(1)}$ . These are unrefereed claims, not independently verified results.

Current status (as of October 2026): Growth claims for the three-, five-, and six-factor cases are reported but unverified; no independently confirmed complete resolution is recorded.

Sources

No solutions have been posted yet.