The Diophantine Equation 8x+py=z2

Abstract

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p≡±3(mod 8), then the equation 8x+py=z2 has no positive integer solutions (x,y,z); (ii) if p≡7(mod 8), then the equation has only the solutions (p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), where q is an odd prime with q≡1(mod 3); (iii) if p≡1(mod 8) and p≠17, then the equation has at most two positive integer solutions (x,y,z).