Büchi's Problem

Büchi's problem, also known as the n squares' problem, is an open problem from number theory named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (x + i)², i = 1, 2, ..., M,... for some integer x. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: 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?

Büchi's problem can be stated in the following way: Does there exist a positive integer M such that the system of equations

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

Since the first difference of the sequence ${\displaystyle \sigma =(x_{n}^{2})_{n=0,\dots ,M-1}}$ is the sequence ${\displaystyle \Delta ^{(1)}(\sigma )=(x_{n+1}^{2}-x_{n}^{2})_{n=0,\dots ,M-2}}$, the second difference of ${\displaystyle \sigma }$ is

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