On the diophantine equation [formula omitted

Abstract

Let D > 2 be a positive integer, and let p be an odd prime not dividing D. In this paper, using the deep result of Bilu, Hanrot and Voutier (i.e., the existence of primitive prime factors of Lucas and Lehmer sequences), by computing Jacobi's symbols and using elementary arguments, we prove that: if ( D , p ) ≠ ( 4 , 5 ) , ( 2 , 5 ) , then the diophantine equation x 2 + D m = p n has at most two positive integer solutions ( x , m , n ) . Moreover, both x 2 + 4 m = 5 n and x 2 + 2 m = 5 n have exactly three positive integer solutions ( x , m , n ) .