Hilbert's tenth problem for systems of diagonal quadratic forms, and B\"{u}chi's problem

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.