In this paper we complete B\"{ u}chi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five squares such that the second differences satisfy \[x_{ k+2}^2 - 2 x_{ k+1}^2 + x_k^2 = 2\] for $k = 1,2,3$, then they must be consecutive. This answers a question of J.~Richard~B\"{ u}chi.
Categories: math.NT math.AG