Problem 467

Prove the following for all large $x$ : there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$ , there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$ .

Source: erdosproblems.com/467

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The only source in hand is the 1980 sentence; no proof, disproof, partial result or proof claim for the site's statement was found in the search whose scope the Current assessment records, and nothing found bears on it beyond its relation to Problems 687 and 689. This is a bounded negative finding, not a certificate of openness.

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. 93, in the chapter of miscellaneous problems. Library home: Erdős and Graham 1980.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; Section 6, item 6, printed p. 108, the remark on $f(x)\ge2$ quoted under Formulation. Library home: Erdos 1980.
ErGr79: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (2) 25 (1979), 325--344. The chapter published in advance of the monograph; its plan (printed p. 326) announces the chapter on covering congruences; the chapter's own covering passages (printed pp. 334--335) concern disjoint coverings of the integers by generalized arithmetic progressions, not covering by residue classes modulo primes. Its library card links this page as a problem-list association; the passage is not there. Library home: Erdős and Graham 1979.

Problem 467

Status

Proof claims

Discussion