Problem 727

../

claims/: The 1 claim page of Problem 727, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $k\geq 2$. Does

$$(n+k)!^2 \mid (2n)!$$

for infinitely many $n$?

Status. Open: the site's label, and its page does not mention the one claim. Johan Land's partial claim of 2026-09-07, a proof for $k=3$ (hence $k=2$) with a Lean development, is pending on the Land claim page. The standing in the frontmatter derives from the claim pages.

Source. erdosproblems.com/727, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #727, https://www.erdosproblems.com/727.

References.

• [Ba29] H. Balakran, On the values of $n$ which make $(2n)!/(n+1)!(n+1)!$ an integer. J. Indian Math. Soc. (1929), 97-100.
• [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\sp{2n}\sb{n})$. Math. Comp. (1975), 83-92.
• [Er68c] P. Erdős, Aufgabe 557. Elemente Math. (1968), 111-113.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1975_prime_factors
• erdos_1975_prime_factors / conjecture_p90_factorial_quotient