Abstract: For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x,y) of the equation y²=px(x²+2). In this paper, using some properties of binary quartic Diophantine equations, we prove that if p≡5 or 7(mod 8), then N(p)=0; ifp≡1(mod 8), then N(p)≤1; ifp>3 and p≡3(mod 8), then N(p)≤2.

Key words:  cubic and quartic Diophantine equation, number of solutions, upper bound