Problem 253

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claims/: The 1 claim page of Problem 253, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1\leq a_1<a_2<\cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i\to 1$. If every infinite arithmetic progression contains infinitely many integers which are the sum of distinct $a_i$ then every sufficiently large integer is the sum of distinct $a_i$.

Status. DISPROVED (LEAN). Cassels's Theorem II ([Ca60], refereed) constructs a sequence with $(c_{n+1}-c_n)/c_n^{1/2+\varepsilon}\to0$, infinitely many elements in every arithmetic progression, and sums of distinct elements of upper density at most $\varepsilon$, so the implication fails; the site records the disproof as Cassels's, and the claim page carries the acceptance. The Lean suffix is the site's label for a 2026 formalization of Cassels's disproof in a public repository, registered by formal-conjectures and the community database, linked on the same claim page, neither built nor audited here.

Source. erdosproblems.com/253, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #253, https://www.erdosproblems.com/253.

References.

• [Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.
• [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Formalization. Statement in formal-conjectures, whose formal_proof attribute (added 2026-09-11, category research solved) names the file src/latest/ErdosProblems/Erdos253.lean of plby/lean-proofs at a pinned commit; Cassels's claim page above links the file at that pin and records what it declares.

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.

• cassels_1960_representation_integers_as_sums_distinct_summands
• cassels_1960_representation_integers_as_sums_distinct_summands / theorem_ii