Erdős Problem #782

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\[a+\left\{ \sum_i \epsilon_ib_i : \epsilon_i\in \{0,1\}\right\}?\]

#782: [BEF90]

number theory

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A question of Brown, Erdős, and Freedman [BEF90]. It is a classical fact that the squares do not contain arithmetic progressions of length $4$.

An affirmative answer to the first question implies an affirmative answer to the second.

Solymosi [So07] conjectured the answer to the second question is no. Cilleruelo and Granville [CiGr07] have observed that the answer to the second question is no conditional on the Bombieri-Lang conjecture.

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