Talk:Perfect number

There are no odd perfect numbers.

There are no such numbers!https://arxiv.org/abs/2101.07176 I am a Green Bee (talk) 10:09, 24 July 2023 (UTC)

@I am a Green Bee: arXiv is not peer reviewed and has lots of false proofs with trivial errors. See WP:ARXIV. PrimeHunter (talk) 13:11, 24 July 2023 (UTC)

Even on arXiv there are levels. Classification as math.GM rather than math.NT suggests that the arXiv mods were not convinced. —David Eppstein (talk) 18:00, 24 July 2023 (UTC)

Just for reference: Saeid-Nia, Hooshang (11 Jan 2021). "There Are No Odd Perfect Numbers". arXiv:2101.07176 [math.GM]. Author's only arXiv paper, and it appears to have sunk without a ripple, so it doesn't look like anyone's convinced. 97.102.205.224 (talk) 21:36, 15 January 2025 (UTC)

rename

The article and the sequence need to be renamed to n-composite numbers, since their prime factorizations do not match well.

examples

6 = 2*3 = squarefree number (A005117(5))

28 = 2²*7 = weak number (A052485(21))

496 = 2⁴*31 = weak number (A052485(460))

8128 = 2⁶*127 = weak number (A052485(7963)) 2A00:6020:A123:8B00:3913:1297:6B6B:CCEF (talk) 13:18, 14 December 2023 (UTC)

You are correct that there is cause for confusion due to the multiple different meanings of perfect. The terms are however standard, and Wikipedia follows the standard terminology. JoshuaZ (talk) 01:02, 17 December 2023 (UTC)

New perfect number found

85921759056 is the new perfect number Rad Deg x! π cos log e tan √ Ans EXP xy ( ) % AC 6 × 1 2 117.55.251.70 (talk) 08:17, 28 May 2024 (UTC)

That number is abundant, not perfect. Also, Wikipedia relies on reliable sources, not original research, so even if it were perfect, we would not be able to include it here until it had been recognized by reliable sources. JoshuaZ (talk) 13:53, 31 May 2024 (UTC)

Another condition

https://www.lirmm.fr/~ochem/opn/opn.pdf is currently cited as reference 21. However, theorem 3, which states that “The largest component of an odd perfect number is greater than 10^62”, is not currently mentioned in the list of conditions that odd perfect numbers must follow.

Whilst I don’t have the mathematical knowledge to understand that article, and hence can not comment on it’s accuracy, I think the fact that it already is referenced means it’s probably a reliable source, and hence I propose adding that theorem to the list of conditions that an odd perfect number must satisfy Person568 (talk) 14:03, 14 October 2024 (UTC)

Actually it is mentioned, I missed it🤦 Person568 (talk) 14:04, 14 October 2024 (UTC)

9 mod 36

This page mentions that “N is of the form N ≡ 1 (mod 12) or N ≡ 117 (mod 468) or N ≡ 81 (mod 324)”. However, the source linked (https://www.austms.org.au/wp-content/uploads/Gazette/2008/Sep08/CommsRoberts.pdf) also mentions that “N must equal 1 mod 12, or 9 mod 36”. Is it worth editing that sentence to mention that (also, am I reading that page correctly)? Person568 (talk) 14:39, 14 October 2024 (UTC)

The purpose of that paper is to sharpen the known result

N is of the form 1 (mod 12) or 9 (mod 36)

to the stronger result

N is of the form 1 (mod 12) or 117 (mod 468) or 81 (mod 324)

Note that both 117 mod 468 and 81 mod 324 are 9 mod 36.

CRGreathouse (t | c) 16:25, 14 October 2024 (UTC)

Impossible inequality for OPN

In section Perfect_number#Odd_perfect_numbers, towards the end, the last condition of "N = q^α ... where ...", it's written

1/q + 1/p1 + ... + 1/pK > ln(K)/(2 ln 2).

They cite [38, 39] but the inequality is in none of these two papers. (On the contrary, they prove the sum must be less than log(2), AFAICS.) This inequality can never hold for any k ≥ 3. Indeed, for k = 3 the LHS is ≤ 1/3 + 1/5 + 1/7 + 1/11 < 0.7671, while the RHS is ln(3)/ln(4) > 0.792. The "best" you can do to get a large value on the LHS for given K is to take all primes up p_{K+1}, but that sum grows slower than that of the odd numbers up to N = 2K+3 which grows like ln(2*K+3)/2, so it can never outperform the RHS which grows like ln(K) / 1.38. — MFH:Talk 14:24, 15 October 2024 (UTC)

This estimate doesn't appear to be included in the sources provided. I notified the contributor responsible for it JoshuaZ on his talk page.--Sapphorain (talk) 20:54, 16 October 2024 (UTC)

I made the version here https://en.wikipedia.org/w/index.php?title=Perfect_number&diff=1212366422&oldid=1211622525 which is I think the correct version. It looks like it got garbled into an incorrect form in this edit https://en.wikipedia.org/w/index.php?title=Perfect_number&diff=next&oldid=1247362157 . If that's correct, we should fix that. JoshuaZ (talk) 21:41, 16 October 2024 (UTC)

I have restored the version prior to that edit. --JBL (talk) 00:14, 17 October 2024 (UTC)

New paper-COI

There's a new paper I wrote on odd perfect numbers https://math.colgate.edu/~integers/z59/z59.pdf . I'm not going to edit the article directly due to the COI issues until/unless others here decide the paper should be cited. There are a bunch of results, some of which are pretty technical and therefore should not be included in the article. However, Proposition 5 may be reasonable to include, but I'm not sure. JoshuaZ (talk) 12:48, 4 July 2025 (UTC)

About 33550336

Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL According to the results so far, the number 33550336 is often used in popular culture. It seems that it is time for us to add these messages to the article. ~2025-40096-62 (talk) 13:25, 11 December 2025 (UTC)

Suggestion for the section Odd perfect numbers

The first sentence of the article reads as follows:

"In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself."

With this definition, it is unclear whether or not the number 1 qualifies as a perfect number.

The section Odd perfect numbers makes no mention of the number 1 at all. I suggest that this issue be addressed directly in this section: It should be stated whether or not 1 qualifies as perfect, and exactly what the reason is. ~2026-13257-12 (talk) 03:01, 5 March 2026 (UTC)

I don't see how it's unclear. The number 1 has no positive divisors excluding the number 1 itself. Therefore the sum of all such divisors is 0, which is not equal to 1, so 1 is not a perfect number. I don't think we need to make any change here. --Trovatore (talk) 03:04, 5 March 2026 (UTC)

Agreed. It is even clearer from the equivalent formulation that n is perfect iff $\sigma (n)=2n$ (which avoids the need to invoke the empty sum). --JBL (talk) 22:40, 5 March 2026 (UTC)

Just as if they were called biperfect numbers?? (Note what the prefix bi- means.) Georgia guy (talk) 23:35, 5 March 2026 (UTC)

Let's not go off into the weeds here. The questions, essentially, were, first, is 1 a perfect number? (Answer: no) and second, should we discuss the status of 1 in the "Odd perfect numbers" section? (Answer: no). Speculation about other terminology that could have existed in some alternative history is outside the scope of this talk page. --Trovatore (talk) 00:34, 6 March 2026 (UTC)

Trovatore, my "Just as if" post was targeted solely at JBL and stands for the statement that JBL's definition shows that another term would be more logical than the one we have; it DOES NOT, in any way, mean that I want to adopt the use of the word "biperfect" in Wikipedia. Georgia guy (talk) 00:42, 6 March 2026 (UTC)

Ok. --JBL (talk) 00:59, 6 March 2026 (UTC)

ugh. 1+2+3=6 6 is perfect and 1 is included. so factors of 1 is 1, and the total is obviously 1. SO WDYM? Numberblock 21 (talk) 17:29, 4 April 2026 (UTC)

Can't you do this?

So the largest prime number? why can't we use it to make it a perfect number? 2p-1((2^p)-1). so its 2^136,279,841-1((2^136,279,841)-1). Numberblock 21 (talk) 17:27, 4 April 2026 (UTC)