Brocard's Problem

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For an integer $n\geq0$, $n!=\prod_{j=1}^{n}j$, with $0!=1$. Determine all pairs $(n,m)\in{\mathbb{Z}}_{\geq0}\times{\mathbb{Z}}$ satisfying $n!+1=m^2$. Changing the sign of $m$ does not give a different value of $n$; one may equivalently classify the solutions with $m\geq0$. The question is a complete classification, rather than the construction of some solutions. [S1] [S2]

Statement recorded in the supplied source.

Find all integer solutions to $n! + 1 = m^2$. [S1]

Short English statement

Which factorials become perfect squares when one is added?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 57 (NT-010), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Bruce C. Berndt and William F. Galway, On the Brocard-Ramanujan Diophantine equation n!+1=m^2, Ramanujan Journal 4 (2000), 41-42,

  \nolinkurl{doi:10.1023/A}

  :1009873805276. https://experts.illinois.edu/en/publications/on-the-brocard-ramanujan-diophantine-equation-n-1-msup2sup/. Bibliographic information from the supplied record.

• S3 Eric W. Weisstein, Brocard's Problem, MathWorld. https://mathworld.wolfram.com/BrocardsProblem.html. Bibliographic information from the supplied record.