The board / number theory / #1142
Erdős Problem #1142
OPENdatabase updated 2026-01-23
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 property that \(n > 2\) and \(n - 2^k\) is prime for all \(k \geq 1\) with \(2^k < n\). Following the OEIS [A039669](https://oeis.org/A039669) convention ("Numbers n > 2 such that ..."), we require \(n > 2\) to exclude the trivial cases \(n \leq 2\), for which the primality condition is vacuously satisfied.
Are there infinitely many \(n > 2\) such that \(n - 2^k\) is prime for all \(k \geq 1\) with \(2^k < n\)? The only known such \(n\) are \(4, 7, 15, 21, 45, 75, 105\) (OEIS [A039669](https://oeis.org/A039669)).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1142.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA039669
Original pageerdosproblems.com/1142
Attempts
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Exact modular covering proves there is no solution for \(2^{128}\le n\le340282371933923199876807076826562457784\); the uniform Erdős problem remains open.
PARTIAL