The Brocard–Ramanujan equation n! + 1 = m²

When is $n!+1$ a perfect square? It is for $n=4$, $5$ and $7$, giving $25$, $121$ and $5041$, and computer searches have found nothing else. Proving that these are the only solutions, or even that there are finitely many, is out of reach of current methods.

Problem

Are the only solutions of

$$n!+1=m^2$$

in positive integers the three pairs $(n,m)=(4,5)$, $(5,11)$ and $(7,71)$?

Equivalently, is $n!=m^2-1=(m-1)(m+1)$ impossible for all other $n$?

Background

The problem was posed by Henri Brocard in articles of 1876 and 1885 and, independently, by Srinivasa Ramanujan in 1913. Erdős and Graham listed it in their 1980 book of problems as an old conjecture that is "almost certainly true" but intractable.

What is known

• Searches. No further solutions exist with $n\le 10^{15}$; the search range grew through work of Berndt and Galway (2000), who reached $10^9$, Matson (2017) and Epstein and Glickman (2020).
• Conditional finiteness. Overholt (1993) showed that a weak form of the $abc$ conjecture implies there are only finitely many solutions.
• Partial unconditional results. Naciri (2025) proved that for each $k\ge2$ there are only finitely many solutions in which $m\pm1$ is $k$-free, and that the three known solutions are the only ones in which $m\pm1$ is $7$-free or a prime power.

What would count as progress

• A fourth solution, which would disprove the conjecture and is checked by direct computation.
• An extension of the exhaustive search beyond $10^{15}$, with published code and a description of the sieve so that the range can be audited.
• An unconditional proof that there are only finitely many solutions, or the removal of further side conditions from the known partial results.
• A formal proof against the Lean statement in formal-conjectures.

References

1. M. Overholt, The Diophantine equation $n!+1=m^2$, Bull. London Math. Soc. 25 (1993), 104. doi:10.1112/blms/25.2.104
2. B. C. Berndt and W. F. Galway, On the Brocard–Ramanujan Diophantine equation $n!+1=m^2$, Ramanujan J. 4 (2000), 41–42. doi:10.1023/A:1009873805276
3. A. M. Naciri, On the Brocard–Ramanujan equation with 7-free integers and prime powers, Integers 25 (2025), A71. doi:10.5281/zenodo.16881781

Source

Imported from Erdős Problems 398, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.