bibkey: caceres2019a308090 authors: Pedro Caceres year: 2019 title: “OEIS A308090, gcd(2^n + n!, 3^n + n!, n+1), with the conjectured primality criterion gcd = n+1 implies n+1 prime” doi: null url: https://oeis.org/A308090 claim: “%N a(n) = gcd(2^n + n!, 3^n + n!, n+1). %C Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.” strata_touched:

• D5/S3/Arith/Congruence/CaceresGcdFactorialPowerPrimeCriterion license: citation-only triage: anchor

OEIS A308090

Pedro Caceres’ 2019 comment proposes the gcd-factorial-power primality criterion for OEIS A308090.

Verified locator

• URL: https://oeis.org/A308090

%N a(n) = gcd(2^n + n!, 3^n + n!, n+1).
%O 1,4
%A _Pedro Caceres_, May 11 2019
%C From observation: For n > 3, if n+1 is prime, then a(n) = n+1.
%C Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.

Readings of 2026-09-16: the OEIS entry still marks the line Conjecture (unsigned, entry dated 2019-05-11) with no proof line; the 2021 Mathar note relates the sequence to A090585 except at n = 2 and is not a proof; OpenAlex "A308090" 0 hits; Math.SE API 0 hits; GitHub code search in google-deepmind/formal-conjectures 0 hits. The prime-divisor argument is elementary; the named criterion itself was not found in these surfaces. A GPT Pro reading of arxiv.org, openalex.org and math.stackexchange.com by A-number and by the defining phrases also found no proof; that reading is reported by the selection seat and is ASSUMED-UNVERIFIED. Historical openness beyond them is ASSUMED-UNVERIFIED.