Problem 429

Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$ , then there exists some $n$ such that $n+a$ is prime for all $a\in A$ ?

Source: erdosproblems.com/429

Number theory

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The status-defining source is Theorem 1 of D. Weisenberg, Sparse admissible sets and a problem of Erdős and Graham, Integers 24 (2024), Article A89 (received 24 June 2024, accepted 20 September 2024, published 9 October 2024; a refereed journal): for every nondecreasing unbounded $f$ there is an admissible $A$ with $|A\cap\{1,\ldots,N\}|\le f(N)$ for all $N$ such that no integer $n$ makes $A+n$ a set of primes. So no sparsity threshold exists and the answer is no. The claim page Weisenberg 2024 records the result as accepted on the refereed publication; the standing in the frontmatter is derived from it. The site's curator is not an independent reviewer of this result: the paper's acknowledgement records that he reviewed an earlier draft and advised the author, so his label is not acceptance evidence. The site records DISPROVED (LEAN); the suffix is a catalog label explained under Formalization and the Lean label below; the two external Lean files have not been built or audited in this corpus. The site's further sentence that a variant of the construction also answers Erdős's squarefree ( $p^2$ ) question in the negative is the site's statement: the paper does not treat squarefree numbers.

We24: Weisenberg, D., Sparse admissible sets and a problem of Erdős and Graham. Integers 24 (2024), Article A89, 4 pp., DOI 10.5281/zenodo.13909172; arXiv:2405.12310 (v1 20 May 2024, v2 21 October 2024; the listing carries the journal reference and the DOI); Conjecture 1 and Theorem 1 on pp. 1--2. Library home: Weisenberg 2024; result page Theorem 1.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980); printed p. 85. Library home: Erdős and Graham 1980.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; printed pp. 111--112; public copy at https://users.renyi.hu/~p_erdos/1980-03.pdf. Library home: Erdos 1980.
GuMu84: Gupta, R. and Ram Murty, M., A remark on Artin's conjecture. Invent. Math. 78 (1984), 127--130; and [HB86] Heath-Brown, D. R., Artin's conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2) 37 (1986), 27--38. The paper's references [3] and [4], cited for the existence of a positive integer that is a primitive root modulo infinitely many primes; not held.

Problem 429

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