# On the Diophantine Equation

### From NZJM

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**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

mailto:B.Martin@math.canterbury.ac.nz

**Abstract** Let *r* be a non-negative integer. We show that the Diophantine equation *x*! = *m**y*^{α} has only finitely many positive-integer solutions (*x*,*y*,*m*,α) with *y*,α2 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.

**Keywords**

**Classification** (MSC2000) Primary: 11D45.

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