The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000668

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000668

Mersenne primes (primes of the form 2^n - 1).
(history; published version)

#488 by Sean A. Irvine at Sat Aug 08 18:44:57 EDT 2026

COMMENTS

The smallest odd divisor of (p-1)! + p + 1 is equal to p if and only if p is a Mersenne prime. - Davide Rotondo, Aug 01 2026

KEYWORD

nonn,nice,hard,changed

STATUS

proposed

approved

#487 by Davide Rotondo at Mon Aug 03 03:08:23 EDT 2026

STATUS

editing

proposed

Discussion

Mon Aug 03

03:24

Joerg Arndt: suggest to block account for wasting our time

03:25

Michel Marcus: revert

Fri Aug 07

10:32

Andrei Zabolotskii: The proof is correct. (As of today, AI generated proofs of this complexity are usually correct.)
But I don't see why this is significant enough to be included.

Sat Aug 08

18:44

Sean A. Irvine: Reverted

#486 by Joerg Arndt at Mon Aug 03 02:58:25 EDT 2026

STATUS

proposed

editing

Discussion

Mon Aug 03

03:08

Davide Rotondo: No, so there in't a real proof?

#485 by Davide Rotondo at Sat Aug 01 13:14:44 EDT 2026

STATUS

editing

proposed

Discussion

Sun Aug 02

09:16

Joerg Arndt: can you prove this?

10:08

Davide Rotondo: Dear Joerg this is Arup Kumar  proof.

By Wilson's Theorem, we know that (p-1)! ≡ -1 (mod p), which implies that p always divides (p-1)! + p + 1. For p to be the smallest odd divisor, no odd prime q < p can divide (p-1)! + p + 1.

Since q < p, q divides (p-1)!, so the expression is congruent to p + 1 (mod q). Therefore, p is the smallest odd divisor if and only if no odd prime q < p divides p + 1.

For a Mersenne prime p = 2^n - 1, we have p + 1 = 2^n. Since 2^n has no odd prime factors, the condition is satisfied for all Mersenne primes. This explains why the sequence of such primes aligns with the Mersenne primes.

Arup Kumar

Mon Aug 03

02:58

Joerg Arndt: Arup Kumar has been banned from the OEIS for very good reasons!
You realize that text in the pink box was AI generated?

#484 by Davide Rotondo at Sat Aug 01 13:14:25 EDT 2026

COMMENTS

The smallest odd divisor of (p-1)! + p + 1 is equal to p if and only if p is a Mersenne prime. - Davide Rotondo, Aug 01 2026

STATUS

approved

editing

#483 by Michael De Vlieger at Tue Jun 23 16:44:10 EDT 2026

STATUS

proposed

approved

#482 by Stefano Spezia at Tue Jun 23 16:06:05 EDT 2026

STATUS

editing

proposed

#481 by Stefano Spezia at Tue Jun 23 15:50:31 EDT 2026

REFERENCES

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 82.

STATUS

approved

editing

#480 by Michael De Vlieger at Fri Jun 12 15:45:37 EDT 2026

STATUS

reviewed

approved

#479 by Michel Marcus at Fri Jun 12 12:50:52 EDT 2026

STATUS

proposed

reviewed

Showing entries 1-10 | older changes