content:mathematics:buechi_s_problem

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# 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 integerMsuch that, for all integersxanda, the quantity (x+n)^{2}+acannot be a square for more thanMconsecutive values ofn, unlessa= 0?

Source : 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 $${\displaystyle x_{n}^{2}=(x_{0}+n)^{2}}$$

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