A New Solution of the Diophantine Equation X2 + 1 = 2Y4 Original Research Article

In this paper we use the method of Thue and Siegel, based on explicit Padé approximations to algebraic functions, to solve the equation given in the title. We present an upper bound for the solutions of the Thue equation X4 − 12X2Y2 + 16XY3 − 4Y4 = N, namely y < 4.233 × 107 × N 1/0.3676. For N = 1 we use this bound to solve the equation X2 + 1 = 2Y4.