Erdős Problem #782

declared status 'open'. Formalized: no. Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\ldots,x_k$ where, for some $d$ and all $1\leq i<k$,$$x_i+d\leq x_{i+1}\leq x_i+d+C.$$Do the squares contain arbitrarily large cubes$$a+\left\{ \sum_i \epsilon_ib_i : \epsilon_i\in \{0,1\}\right\}?$$ Prize: no. Tags: number theory.

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