The board / number theory / #1113
Erdős Problem #1113
Sierpinski numbers
OPENdatabase updated 2025-12-28
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).
A *covering set* for a positive odd integer \(k\) is a finite set of primes \(P\) such that every number of the form \(k \cdot 2^n + 1\) is divisible by at least one prime in \(P\).
**Erdős Problem 1113.** Do there exist Sierpiński numbers that possess no finite covering set of primes? Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply that there are infinitely many Fermat primes.
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/1113.Prizenone
Topicsnumber theory, covering systems
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA076336
Original pageerdosproblems.com/1113
Attempts
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The Izotov candidate is re-proved Sierpiński; any finite cover is certified to require at least 686 primes, including one at least 376843822247957 with period at least 94210955561989, but unbounded le
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