Prove that for all nonnegative integers a and b, such that 2a² + 1 = b², there are two nonnegative integers c and d such that 2c² + 1 = d² and a + c + d = b, or give a counterexample.

(For example if a = 0 and b = 1, or a = 2 and b = 3 then c=0 and d = 1.)