Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 157

claims/: The 2 claim pages of Problem 157, one per claimant's result; the problem's standing derives from them.

Statement. Does there exist an infinite Sidon set which is an asymptotic basis of order 3?

Status. Proved, the site's label; the site's curator credits Pilatte. The standing rests on two accepted claim pages: Pilatte 2023, the construction refereed in Compositio Mathematica (2024), and [[problems/additive_bases/E0157/claims/2026_08_25_alexeev|an AI-generated elementary proof]], first posted as a Lean development on 2026-08-25 and registered on the site's proof-claims page on 2026-09-15 with a write-up, whose Lean theorem, in an earlier form of the construction, this corpus built and audited.

Source. erdosproblems.com/157, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #157, https://www.erdosproblems.com/157.

References.

[Pi23] Pilatte, C., A solution to the Erdős-Sárközy-Sós problem on asymptotic Sidon bases of order 3. arXiv:2303.09659 (2023).

Formalization. The site records none, and formal-conjectures has no statement file for the problem. Boris Alexeev's lean-proofs repository proves the statement as Erdos157.erdos_157, by an earlier form of the elementary construction over the field of $2^{1024}$ elements; this corpus built and audited that theorem, as its claim page records. The write-up's version over $\mathbb{F}_2$ , Erdos157.Binary.erdos_157, was not built here.

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Linked library material

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Linked from (10)

problems/additive_bases/E0157/claims Additive Bases and Sidon Sets additive_bases/cilleruelo_2015_sidon_sets_asymptotic_bases Theorem 1.1 (p. 1): for all large N, Z_N has a Sidon set that is a basis of order 3 Theorem 1.2 (p. 2): some B_2[2] sequence of positive integers is an asymptotic basis of order 3 Theorem 1.3 (p. 2): for every ε > 0 there is a Sidon basis of order 3 + ε additive_bases/pilatte_2023_solution_erdos_sarkozy_sos_problem_asymptotic Theorem 5.4 (p. 11): almost surely the random modified Cilleruelo set is a Sidon asymptotic basis of order 3 additive_bases/pliego_2024_erdos_turan_conjecture_growth_b_2 Conjecture 1.2 (p. 4): for every eps > 0 a Sidon sequence in which every large n is a_1 + a_2 + a_3 with a_3 <= n^eps

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