On the diophantine equation x2+Dm=pn

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 Jacobiʹs symbols and using elementary arguments, we prove that: if (D,p)≠(4,5),(2,5), then the diophantine equation x2+Dm=pn has at most two positive integer solutions (x,m,n). Moreover, both x2+4m=5n and x2+2m=5n have exactly three positive integer solutions (x,m,n).