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

Updated Oct 8

Problem 16

claims/: The 1 claim page of Problem 16, one per claimant's result; the problem's standing derives from them.

Statement. Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$ ?

Status. DISPROVED (LEAN), the site's label (page last edited 05 April 2026): Chen's 2023 preprint [Ch23] shows that the odd integers not of the form $2^k+p$ are not a finite union of arithmetic progressions plus a set of density zero, so the answer is no; the site's Lean marker corresponds to the proof the formal-conjectures catalog links, Daniel Chin's Lean file, which proves that no single infinite progression plus a progression-free remainder equals that set of odd integers, a statement that implies the negative answer. The claim page Chen's disproof records the acceptance evidence: the curator of erdosproblems.com, Thomas Bloom, with no refereed publication found; the corpus has not built or audited the Lean proof.

Source. erdosproblems.com/16, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #16, https://www.erdosproblems.com/16.

References.

[Ch23] Chen, Y.-G., A conjecture of Erdős on $p+2^k$ . arXiv:2312.04120 (2023).
[Er50] Erdős, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.
[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.

Formalization. Statement in formal-conjectures, at the file's revision of 2026-09-18: tagged research solved with answer(False) and linking Daniel Chin's Lean proof ErdosProblem16 in https://github.com/danielchin/proofs (announced 2026-02-25 as written with Gemini 3.1 Pro and Antigravity; the catalog's docstring instead credits Chin using Aristotle, and the author's own statement is followed here). The corpus has not built or audited the proof, so the label supplies no formal-verification credit; the claim page gives the pinned link.

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Linked from (15)

Chen's disproof of the progression-plus-null-set conjecture problems/additive_bases/E0016/claims Additive Bases and Sidon Sets Additive Bases and Sidon Sets additive_bases/romanoff_1934_uber_einige_satze_der_additiven Satz II (p. 668): the integers that are a prime plus a power of a have positive lower density primes/chen_2023_conjecture_erdos_p_2_k Theorem 1.1: no density-zero perturbation of the non-representable odd integers is finitely many progressions plus a null set Theorem 1.15: progressions in the non-representable odd integers can be taken with squarefree-times-power-of-two modulus Theorem 1.3: the least modulus of a progression lying in the non-representable odd integers up to density zero is 11184810 Theorem 1.5: the 48 residues a for which 11184810h + a is a longest quasi-non-representable progression Theorem 1.9: a non-representable odd integer lies in a non-representable progression iff some m > 1 meets every a - 2^k Theorem 3.1: Erdős's Conjecture A on the odd integers not of the form p + 2^k is false primes/erdos_1950_integers_form_related_problems Theorem 3 (p. 113): an arithmetic progression of odd numbers containing no integer 2^k + p

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