The board / number theory / #237
Erdős Problem #237
PROVED LEANdatabase updated 2026-03-04
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`f A n` is the number of solutions to \(n = p + a\) with \(p\) prime and \(a \in A\). A solution is determined by \(a\), so this counts the \(a \in A\) with \(a \leq n\) and \(n - a\) prime.
Let \(A\subseteq \mathbb{N}\) be a set such that \(\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N\) for all large \(N\). Let \(f(n)\) count the number of solutions to \(n=p+a\) for \(p\) prime and \(a\in A\). Is it true that \(\limsup f(n)=\infty\)? The answer is yes, as proved by Chen and Ding [ChDi22]. Here `A` may contain \(0\). This changes each \(f(n)\) by at most one, so it does not affect the answer.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/237.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/237
Attempts
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