IJPAM: Volume 26, No. 4 (2006)

A NOTE ON THE DIOPHANTINE EQUATION $x! + A = y^2$

Alain Togbé
Department of Mathematics
Purdue University North Central
1401 S, U.S. 421, Westville IN 46391, USA
e-mail: atogbe@pnc.edu


Abstract.In this paper, we study the variant of the Brocard-Ramanujan Diophantine equation $x! + A = y^2$. In fact, we consider the cases $A=4k+2$, $A=4(k+1)+2$, $A=4(k+1)+3$, where $k\in \NN$. We prove that the simultaneous equations have a unique integral solution if and only if $k+2$ is a square.

Received: February 6, 2006

AMS Subject Classification: 11D85, 11Y50

Key Words and Phrases: Brocard-Ramanujan, Diophantine equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 26
Issue: 4