Problem Statement
Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$ satisfies at least two of the congruences $\equiv a_p\pmod{p}$?
Categories:
Number Theory
Progress
One can ask a similar question replacing $2$ by any fixed integer $r$ (provided $n$ is sufficiently large depending on $r$).See also [687] and [688].
This problem (with $2$ replaced by $10$) is Problem 45 on Green's open problems list.
Source: erdosproblems.com/689 | Last verified: January 16, 2026