The board / number theory / #1209
Erdős Problem #1209
OPENdatabase updated 2026-04-04
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The counterexample sequence: `a 0` is a prime at least `f 0`, and `a (k + 1)` is a prime larger than `a k` and `f (k + 1)` with `(k + 1) + a (k + 1)` composite.
Let \(A=\{a_1<a_2<\cdots\}\) be a sequence of integers which tends to infinity sufficiently fast. If there is an \(n\) such that all \(n+a_k\) are primes then must there exist infinitely many such \(n\)? Erdős [Er80] wrote 'unless I overlook a trivial way of getting a counterexample these questions are quite hopeless'. There is indeed a trivial counterexample (a variant of the construction in [erdosproblems.com/429]): define \(a_1=2\) and for \(k\geq 2\) let \(a_k>a_{k-1}\) be a prime such that \(a_k+k\equiv 0\pmod{q_k}\), where \(q_k\) is some prime not dividing \(k\). This sequence can be made to grow arbitrarily fast See also [erdosproblems.com/429] and [erdosproblems.com/1102].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1209.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1209
Attempts
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[D] Every integer \(|n|\le10000\) except \(n=\pm1\) is explicitly ruled out for the always-squarefree variant by a prime-square witness with \(p\le1423\) and \(k\le250270\); within this range existenc
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