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 =2 y2 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.