Problem 782

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claims/: The 1 claim page of Problem 782, one per claimant's result; the problem's standing derives from them.

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Statement. 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

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

Status. 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.

Source. erdosproblems.com/782, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #782, https://www.erdosproblems.com/782.

References.

• [BEF90] Brown, T. C. and Erdős, P. and Freedman, A. R., Quasi-progressions and descending waves. J. Combin. Theory Ser. A (1990), 81-95.
• [CiGr07] Cilleruelo, Javier and Granville, Andrew, Lattice points on circles, squares in arithmetic progressions and sumsets of squares. (2007), 241-262.
• [So07] Solymosi, József, Elementary additive combinatorics. (2007), 29-38.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• brown_1990_quasi_progressions_descending_waves
• brown_1990_quasi_progressions_descending_waves / definition_p2
• brown_1990_quasi_progressions_descending_waves / question_p12
• brown_1990_quasi_progressions_descending_waves / theorem_1
• cilleruelo_2007_lattice_points_circles_squares_arithmetic_progressions
• cilleruelo_2007_lattice_points_circles_squares_arithmetic_progressions / conjecture_6