ARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY OF THE DIOPHANTINE EQUATION 8x4 +1= y2

Abstract

There is no arithmetic progression consisting of square terms and with

a square common difference. Alternatively, the diophantine equation

1 + x4 = 2y2 has no solution in positive integers. Consequently, the

diophantine equation 8x4 +1 = y2 has no positive integral solution other

than x = 1, y = 3, a clear indication that no balancing number other

that 1 is a perfect square