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Erdős Problem #10
OPENdatabase updated 2025-08-31
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 set of natural numbers that can be written as a sum of a prime and at most \(k\) powers of \(2\).
Is there some \(k\) such that every large integer is the sum of a prime and at most \(k\) powers of \(2\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/10.Prizenone
Topicsnumber theory, additive basis, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA387053
Original pageerdosproblems.com/10
Attempts
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exact exhaustive table through 1,117,175,146; proved computationally that 1,117,175,146 is the first integer requiring four powers, while the uniform boundedness question remains open.
PARTIAL