The board / number theory / #688
Erdős Problem #688
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Define \(\epsilon_n\) to be maximal such that there exists some choice of congruence class \(a_p\) for all primes \(n^{\epsilon_n} < p \leq n\) such that every integer in \([1,n]\) satisfies at least one of the congruences \(\equiv a_p \pmod p\).
Estimate \(\epsilon_n\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/688.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/688
Attempts
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Proved the exact Jacobsthal-quotient reduction, the unconditional bound \(\limsup\epsilon_n\le e^{-1}\), and a from-scratch exact table \(L(n)\) for every \(11\le n\le70\); the missing uniform linear
PARTIAL