====== Büchi's problem ======

//Büchi's problem//, also known as the //n squares' problem//, is an open ( i.e. as yet unsolved ) problem in number theory named after the Swiss mathematician Julius Richard Büchi (1924-1984).

It can be stated as :

>Does there exist a positive integer //M// such that, for all integers //x// and //a//, the quantity (//x// + //n//)<sup>2</sup> + //a// cannot be a square for more than //M// consecutive values of //n//, unless //a// = 0?
>\\ Source : [[https://en.wikipedia.org/wiki/B%C3%BCchi%27s_problem|Wikipedia]]

Or, as a formula :

Does

$${\displaystyle {\begin{cases}x_{2}^{2}-2x_{1}^{2}+x_{0}^{2}=2\\x_{3}^{2}-2x_{2}^{2}+x_{1}^{2}=2\\{}\quad \vdots \\x_{M-1}^{2}-2x_{M-2}^{2}+x_{M-3}^{2}=2\end{cases}}}$$

only have solutions satisfying $${\displaystyle x_{n}^{2}=(x_{0}+n)^{2}}$$

?