Number Theory

    

Solution of the Diophantine Brocard Equation

Authors: Kurmet Sultan

It is shown that Brocard’s Diophantine equation can have a solution only if the factorial is represented as a product of two natural numbers that differ by 2, or as a product of four consecutive natural numbers. Then a theorem was proven stating that the product of m consecutive natural numbers cannot be represented as a product of two natural numbers differing by 2 if m≠4. After this, it was proven that it is impossible to represent a factorial greater than 7! in the form of the product of four consecutive natural numbers and two natural numbers differing by 2, it follows that Brocard’s equation has no solutions, with the exception of the well-known three factorials.

Comments: 9 Pages.

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Submission history

[v1] 2021-03-06 20:20:29
[v2] 2021-09-06 22:40:29
[v3] 2021-09-11 01:48:36
[v4] 2024-02-27 03:50:01

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