Talk:Mersenne prime

All Mersenne numbers with the prime exponents not only primes pass the Fermat primality test for a = 2.

The part of the article starting with the following sentence

"All Mersenne numbers with the prime exponents not only primes pass the Fermat primality test for a = 2"

is own work of Mr. Matt Kalinski (possibly the Wikipedia user with the Mattedia nick). It is not confirmed by any other RS.

Is it allowed to add this kind of edits to the Wikipedia?

Regards Szelma W (talk) 11:01, 24 September 2023 (UTC)

Incorrect Statements in the Theorems about Mersenne numbers section since Dec 7 2013?

In the Theorems about Mersenne numbers section of the Mersenne Prime page I believe there is incorrect information. The statements I am referring to are part of the 3rd bullet point the discusses the form for factors of Mersenne numbers. (I am not saying the main point of this section is wrong, just there is an incorrect statement made.) Specifically, the sentence I believe to be incorrect is:

"As a result, for all positive integers x, q is a factor of 2^x − 1 if and only if p is a factor of x."

If I understand the statement correctly, then counter-examples are easy & plentiful. For example, a counter-example would be:

Let x = 5*7 = 35. (So the only factors of x are 5 and 7.)

Then q = 71 is a factor of 2^x − 1. (2³⁵ − 1 = 0 mod 71)

So for the "if and only if" statement to be correct, then q (or 71) needs to be a factor of 2⁵ − 1 or 2⁷ − 1, but neither of those are true. Indeed, you can easily check all smaller x, and find that the smallest x where q=71 is a factor of 2^x − 1 is x=35.

I checked the edit history and see that this information has been a part of the article since Dec 7, 2013 -- added by @User:Blackbombchu. So I am surprised it could be incorrect and exist on the page for so long. (Which makes me question if I am correct about calling this information incorrect, but it certainly seems incorrect to me.)

(I am not sure if this is the appropriate place to make this report of incorrect information, if there is a more appropriate way to report this suspected mistake please let me know.) Ssebeny (talk) 20:56, 16 September 2024 (UTC)

What is p supposed to be in your example? (Whether or not you're right, it is certainly a problem that the section in question consists entirely of OR by the various people who wrote it, with no reliable sources.) --JBL (talk) 01:04, 17 September 2024 (UTC)

For the counter-example I provide, x=35 so the only factors of x are 5 & 7. You could take p=5 & c=7, or p=7 & c=5. Either way, q=71 is not a factor of 2⁵ − 1 or 2⁷ − 1 and is a factor of 2³⁵ − 1, which seems to violate the if and only if portion of the statement in the article at a minimum. Ssebeny (talk) 03:35, 17 September 2024 (UTC)

Hi! I do not understand your doubts. The first sentence of the theorem says: "If p is an odd prime, then every prime q that divides 2^p − 1"

In your example q=71 is not a divisor of 2^p − 1 either for p=5 or p=7

The sentence "As a result, for all positive integers x, q is a factor of 2^x − 1 if and only if p is a factor of x." is also correct as given in your example q=71 is the factor of 2^x − 1 for x=35 and p=5 or p=7 are factors of x=35. Regards

Szelma W (talk) 12:40, 17 September 2024 (UTC)

Section "Recent development" is irrelevant (and self-promotional)

This section is identical to what the same user keeps inserting into https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test - and in both cases this material is irrelevant (and self-promotional). Suggest prompt removal as well as a serious editorial/admin discussion with that user. Serge Batalov (talk) 21:11, 16 October 2024 (UTC)

When I explain that the material is irrelevant, the user resorts to blames of discrimination as well as my incompetence. And both are untrue - I am discussing strictly the text meterial of the section and its suitability for both Wiki articles. Serge Batalov (talk) 21:16, 16 October 2024 (UTC)

Reference 40 seems irrelevant.

40. Zalnezhad et al. leads to a vacuous unpublished preprint that has no connection to the subject. Suggest removal. Serge Batalov (talk) 06:58, 22 October 2024 (UTC)

I agree and have removed it. --JBL (talk) 23:56, 22 October 2024 (UTC)

Wiki Education assignment: Number Theory I

This article was the subject of a Wiki Education Foundation-supported course assignment, between 10 September 2024 and 15 December 2024. Further details are available on the course page. Student editor(s): Aabratcher98 (article contribs).

— Assignment last updated by Aabratcher98 (talk) 17:53, 10 December 2024 (UTC)

"2^61-1" listed at Redirects for discussion

The redirect 2^61-1 has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 January 24 § 2^61-1 until a consensus is reached. Utopes _{(talk / cont)} 06:25, 24 January 2025 (UTC)

Factoring composite Mersenne numbers

@Szelma W: As mentioned in the introduction of this article, the term "Mersenne number" is ambiguous: some authors use it to mean the numbers that are 1 less than a power of 2, while others use it to mean numbers that are one less than a power of 2 with prime exponent. I think the section Mersenne_prime#Factorization_of_composite_Mersenne_numbers is entirely written with the second convention. In that case I do not think the statement "most of the Mersenne numbers are not primes" can be justified by current knowledge. After your recent edits this convention is now established right at the end, but I believe that the convention is in fact active throughout the section. --JBL (talk) 17:43, 5 March 2025 (UTC)

I'm new to this game, but I just noticed that the first paragraph defines Mersenne Primes in two different ways. I believe the first sentence should be changed to indicate that Mersenne Primes are of the form two to the n power minus 1, where n is also prime. The requirement that n be prime is left out of the definition in the first sentence, but included in the last sentence of that same paragraph. This is confusing and should be changed.

I note that the definition given at the bottom of the first GIMPS page also leaves out the prime requirement on n, but adds it in their glossary. I suspect there is general agreement that n be prime, but I will leave it to others here to make that correction to the first paragraph here and perhaps also to the wording at the bottom of the GIMPS home page. Jadney (talk) 18:04, 8 December 2025 (UTC)

The definition of Mersenne prime is "a prime number that is one less than a power of 2". It is a theorem that for each Mersenne prime, the exponent of 2 is also a prime number, but this is not part of the definition. --JBL (talk) 18:12, 8 December 2025 (UTC)

Okay, thanks for the clarification. In that case, the last sentence of the first paragraph should be changed or removed.

This also means that the definition at the bottom of the GIMPS home page is correct, but their glossary definition is not. Do you have clout on GIMPS, so you could fix that also? Jadney (talk) 18:46, 8 December 2025 (UTC)

No, the last sentence of the first paragraph is correct: let S be the set of numbers that are one less than a power of 2 and also prime ("Mersenne primes"). Theorem: if 2^m - 1 is in s, then m is prime. Corollary: Let T be the set of prime numbers of the form 2^p - 1 where p is prime; then S = T. The last sentence of the first paragraph is the (true) corollary: since S = T, we could just as well use the definition of T instead of the definition of S. I have no involvement with GIMPS, but I also do not think there are any errors here that need correcting. --JBL (talk) 18:52, 8 December 2025 (UTC)

"Theorems about Mersenne Numbers" mis-named

This page has a section entitled "Theorems about Mersenne numbers", which contains ten theorems.

But theorems 1, 2, 3, 4, and 7 don't mention Mersenne numbers or Mersenne primes. That's five out of ten!

So this section needs fixing, in one of the following ways:

1. Remove the five theorems that don't mention Mersenne, or
2. Change the section name, or
3. Rephrase the text so that the connection between theorems 1,2,3,4,7 and Mersenne becomes clear.

I understand that theorems 1, 2, and 3 are prerequisites for a rigorous treatment of Mersenne primes. That doesn't justify their inclusion here. The axioms of arithmetic and set theory are also prerequisites, and they aren't listed here. — Lawrence King ^(talk) 04:12, 18 August 2025 (UTC)

All of the five theorems you mention assert something about numbers of the form $2^{n}-1$, i.e., about Mersenne numbers. --JBL (talk) 19:05, 18 August 2025 (UTC)

Question about: Mp is prime for only 43 of the first two million prime numbers (up to 32,452,843)

What does the number 32,452,843 mean? If it's just a number, then it is less than 2power27. So there cannot be 43 Mersenne numbers up to 32 million. Did I miss something ? Hexagone59 (talk) 19:06, 14 October 2025 (UTC)

32,452,843 is the 2,000,000th prime number. It refers to the prime numbers that go in the exponent. –LaundryPizza03 (dc̄) 19:10, 14 October 2025 (UTC)

Thx Hexagone59 (talk) 19:16, 14 October 2025 (UTC)

Semi-protected edit request on 1 January 2026

On the line that is right above the table that shows the prime factorization of composite mersenne numbers whose exponent is prime, the sequence https://oeis.org/A244453 is linked, but it should be https://oeis.org/A054723. ChessCoder (talk) 23:44, 1 January 2026 (UTC)

 Not done The OEIS sequence mentioned gives the factorizations (the last column) and prominently links the exponents (the first column); this is much more useful than linking the sequence of exponents (which does not provide the factorizations nor link to it prominently). --JBL (talk) 00:03, 2 January 2026 (UTC)