Number Theory

    

Solution of the Brocard's Problem

Authors: Kurmet Sultan

It was proved that for a factorial to be a solution to Brocard's problem, it must be representable as a product of two natural numbers differing by 2. It was then proved that only factorials, which are known solutions to Brocard's problem, can be represented as a product of two natural numbers differing by 2. It follows from this that Brocard's Diophantine equation n!=t^2-1 has no solutions other than the classical one (n,t) = (4,5), (5,11), (7,71).

Comments: 3 Pages.

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Submission history

[v1] 2023-04-17 16:46:33
[v2] 2023-12-11 20:47:23
[v3] 2023-12-29 01:31:57
[v4] 2026-02-03 21:10:11

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