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\}?

Source: erdosproblems.com/782

Number theory

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Status

Open

No claim settles this problem.

Open; the site labels the problem OPEN. Under the Bombieri-Lang conjecture both questions have a negative answer, by the conditional result of Cilleruelo and Granville.

Problem 782

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Proof claims

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