# On the Diophantine Equation

### New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 65-66

Ben Martin

Department of Mathematics and Statistics

University of Canterbury

Private Bag 4800

Christchurch 1

NEW ZEALAND

Abstract Let r be a non-negative integer. We show that the Diophantine equation x! = myα has only finitely many positive-integer solutions (x,y,m,α) with y$\geq$2 and m having no more than r distinct prime factors, and we list these solutions explicitly when m is a prime power. This generalises results of Tonien, who considered the case when m is prime.

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Classification (MSC2000) Primary: 11D45.