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Indexed under : Mathematics

Wikenigma - an Encyclopedia of Unknowns Wikenigma - an Encyclopedia of the Unknown

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)2 + a cannot be a square for more than M consecutive values of n, unless a = 0?

Source : Wikipedia

Or, as a formula :


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

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


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