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Addition for the odd perfect number and COI- (III?)

I have a recent paper with Sean Bibby and Pieter Vyncke where we prove that an odd perfect number ${\displaystyle N}$ with third largest prime factor ${\displaystyle a}$ must satisfy ${\displaystyle a<2N^{\frac {1}{6}}.}$ paper here(pdf). Since I'm an author, there's an obvious COI issue. I'm also just not sure that this should be included or not. The current version of the article has a lower bound on the third largest prime factor, but not upper bound. (I'm not aware of a non-trivial upper bound in the literature prior to our work, but our upper bound is pretty weak.) Should this result be included in the section? JoshuaZ (talk) 12:39, 6 December 2021 (UTC)

Only one of the three largest primes can be the special prime, and so the best case is that two have exponent 2 and the third has exponent 1. That gives ${\displaystyle a<N^{\frac {1}{5}}.}$ This is a respectable improvement over that naive exponent, and effective to boot. The only other information we have on a, to my knowledge, is Iannucci's 20+ year old lower bound ${\displaystyle a\geq 101.}$ So I think this is worth including. Joshua, are you aware of similar upper bounds for other prime factors? I believe my argument generalizes, with the n-th largest prime factor ${\displaystyle p_{n}}$ having the trivial bound ${\displaystyle p_{n}<N^{\frac {1}{2n-1}}.}$ I'm not aware of any aside from yours, but I haven't been following OPNs closely for a while.

I'll give the paper a look and give it a go later today if I have a chance.

Disclaimer: I'm an admin who has asked to be notified in cases like this where authors have work relevant to this page but are wary of COI concerns.

- CRGreathouse (t | c) 17:50, 6 December 2021 (UTC)

Oh yes, you cite bounds for the largest and second-largest primes -- I think those should be added to the article as well. The combined bounds like your bound on the product abc are also interesting to me but should probably be left out of a general-interest article like this. A more focused OPN article written for specialists would certainly cover these in some consistent way but that's not the way we're organized at the moment. - CRGreathouse (t | c) 17:55, 6 December 2021 (UTC)

The best bounds for the largest and second largest (both upper and lower) are actually already in the article. I agree that the product abc bound should not be included in this article. (For the same reason there was a product bc bound which we also haven't included in this article.). JoshuaZ (talk) 18:59, 6 December 2021 (UTC)

 Done Perfect, makes my life easier! - CRGreathouse (t | c) 19:08, 6 December 2021 (UTC)

In : def ffs(x):
     ...:   x = gmpy2.mpz(x)
     ...:   return gmpy2.bit_length(x&-x)-1

     ...: def extractoddfactor(N):
     ...:   return N//(2**ffs(N))

In : def checkifperfectnum(N):
     ...:    a = ffs(N)
     ...:    e = extractoddfactor(N)
     ...:    ex = 2**(a+1)-1
     ...:    if e == ex: return True
     ...:    else: return False

In : a = pow(1, 3)
     ...: for x in range (3,8192,2):
     ...:    a += pow(x,3)
     ...:    b = extractoddfactor(a)
     ...:    if checkifperfectnum(a):
     ...:         print (a,b, gmpy2.is_prime(b), checkifperfectnum(a))
     ...: 
     ...: 

28 7 True True
496 31 True True
8128 127 True True
130816 511 False True
2096128 2047 False True
33550336 8191 True True
536854528 32767 False True
8589869056 131071 True True
137438691328 524287 True True
2199022206976 2097151 False True
35184367894528 8388607 False True
562949936644096 33554431 False True

climb=1*4-1
loop:
  n=((3*climb-2)*(3*climb-1))//2
  climb=climb*4-1

In : def altpnusewithnumbertopower(N, withstats=False):
     ...:    N = 2**N-1
     ...:    if withstats==True:
     ...:      print(f"Answer = pow({N},3) + -{N} * pow({N},2) + (({N}+1)//2) * {N} + 0")
     ...:      print(f"Components: pow(N,3) = {pow(N,3)},  -N:  -{N}, pow(N,2) = {pow(N,2)}, ((N+1)//2) = {((N+1)//2)}, N = {N}, 0")
     ...:    return pow(N,3) + -N * pow(N,2) + ((N+1)//2) * N + 0
     ...: 

In : for x in range(2,16):
     ...:     print (altpnusewithnumbertopower(x))
     ...: 
6
28
120
496
2016
8128
32640
130816
523776
2096128
8386560
33550336
134209536
536854528

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)

New paper by Clayton and Hansen

There is a new paper by Graeme Clayon and Cody Hansen in Integers which improves upon the prior linear bounds relating the total number of distinct prime factors to the total number of prime factors of an odd perfect number. If no one objects, I will replace my bound with their bound since their bound is better for all values of $k$. JoshuaZ (talk) 18:24, 27 November 2023 (UTC)

Properly published, so ok to use. I see no reason to object. —David Eppstein (talk) 19:01, 27 November 2023 (UTC)

Ditto, go for it. --JBL (talk) 19:04, 27 November 2023 (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 = 2*7 = weak number (A052485(21))

496 = 2*31 = weak number (A052485(460))

8128 = 2*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 Misplaced Pages 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, Misplaced Pages 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 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/search/?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/search/?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)

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