Talk:Perfect number

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The Next Perfect Number is 28 = 1 + 2 + 4 + 7 + 14[edit]

I added... The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next paragraph is confusing with explaining how 56 = 28 x 2. 2601:580:107:4A57:6D32:859F:D592:8ABB (talk) 11:24, 6 July 2019 (UTC)[reply]

The explanation given in the next paragraph of the lede is perfectly clear (provided of course that one takes the pain of reading the whole sentence...) Sapphorain (talk) 11:32, 6 July 2019 (UTC)[reply]

Hi there. The next paragraph is intended to illustrate a condition for perfect numbers in terms of the divisor function. The first paragraph is intended to be as accessible as possible to readers, but the divisor function is an important concept in number theory so it is helpful to mention it in the next paragraph. — Bilorv (he/him) _(talk) 11:38, 6 July 2019 (UTC)[reply]

Gallardo's Result[edit]

Is Gallardo's result (in Minor results) true? In the linked paper he implicitly assumed that $x+a$ and $x^{2}-ax+a^{2}$ are coprime, but it might not be the case if both $a$ and $x$ are even. 219.78.80.30 (talk) 06:48, 2 September 2020 (UTC)[reply]

That's a good point. I don't see how he's getting that step either. Maybe raise this on Mathoverflow or contact him directly. JoshuaZ (talk) 22:25, 10 September 2020 (UTC)[reply]

Changes to odd perfect number section and COI[edit]

Same issue as before but another paper. Again, I'm the author so I have a clear COI, so this needs to be okayed before I make any edits. This paper has multiple possibly relevant inequalities. Paper is here.

First, the page currently cites Grun's bound that the smallest prime factor must be less than ${\frac {2k+8}{3}}.$ The paper has much better than linear bounds in general, but those bounds are long and technical, and so probably shouldn't be on this page by themselves. However, Corollary 4 on page 43, is equivalent to in the notation on this page that the smallest prime is at most ${\frac {1}{2}}k-{\frac {1}{2}}$ which is tighter than Grun's bound. Should that be included? My inclination is to include that bound but *not* the more technical non-linear bounds.

Second, the page currently has the bound that $\alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq (21k-18)/8$ . This paper improves that bound to $\alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {66x-191}{25}}.$ This is stronger when $k\geq 9$ and thus for all odd perfect numbers. It would make sense to include this tighter bound.

Are there objections to making these two changes? JoshuaZ (talk) 01:45, 3 August 2021 (UTC)[reply]

Joshua, is the latter supposed to have k rather than x? In that case yes, let's include both. Otherwise, what is x in this context?

P.S. Feel free to ping me if this sort of question comes up in the future, the OPN community is pretty small (not that I'm a member, but I dabble, and I don't know of any other Wikipedians who do more than the two of us in the area). - CRGreathouse (t | c) 18:21, 5 August 2021 (UTC)[reply]

Sorry, yes, that should have read

{\frac {66k-191}{25}}.

I'll wait another day then and if no one has any objections, I'll make these changes. JoshuaZ (talk) 20:01, 5 August 2021 (UTC)[reply]

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

I have a recent paper with Sean Bibby and Pieter Vyncke where we prove that an odd perfect number $N$ with third largest prime factor $a$ must satisfy $a<2N^{\frac {1}{6}}.$ 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)[reply]

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

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

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

p_{n}

having the trivial bound

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)[reply]

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)[reply]

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)[reply]

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

Three closed discussion threads. Please do not post original research to this talk page. It should be only for discussion of how to use published sources to improve our article. Additional messages of this sort may be removed altogether. —David Eppstein (talk) 01:16, 8 May 2022 (UTC)[reply]

The following discussion has been closed. Please do not modify it.

conjecture regarding the divisors of Perfect Numbers (PN)[edit]

I conjecture that the product of the divisors a PN derived from 2^p-1(2^p - 1) will equal PN^p-1. For instances: 1×2×4×7×14 = 28² and 1×2×4×8×16×31×62×124×248 = 496⁴. Unfortunately, I cannot prove this. Also, in the first several such PN, there is only 1 odd divisor (> 1) and which is a prime number (the first several are 3, 7, 31, 127, 8191 and 131071)--does this persist? Wmsears (talk) 01:54, 26 December 2021 (UTC)[reply]

This will be true in general. It follows from a more general theorem that the geometric mean of the divisors of a positive integer is exactly the square root of the number. However, your observation, and the observation that this would follow from this are both Original research and therefore not suitable for Wikipedia. In the future, if you have similar math questions, I suggest checking out Math Stack Exchange. JoshuaZ (talk) 02:39, 26 December 2021 (UTC)[reply]

@Wmsears: Note that the term divisor includes the number itself so their product becomes PN^p. The divisors of a number can be listed in pairs like 28 = 1×28 = 2×14 = 4×7 (a square n² also has one unpaired divisor n). Your formula can be worked out from this by considering the number of divisors. Hint: They are all of form 2^m or 2^m×(2^p - 1). The divisors without the number itself are called the proper divisors.

Regarding your other observation, it's known that 2^p-1(2^p - 1) is a perfect number if and only if 2^p - 1 is prime (called a Mersenne prime). The only odd divisors of 2^p-1(2^p - 1) are 1 and 2^p - 1. Your observation follows from this so it persists. PrimeHunter (talk) 04:47, 26 December 2021 (UTC)[reply]

I think the Odd Cubes section could be made more accurate[edit]

For odd cubes to work they only seem to work for every second even perfect number, I don't think the article makes that clear, unless I'm miss reading the explanation ( which I may be but since I didn't quite understand it maybe it could be made clearer):

In [3385]: def ffs(x):
     ...:   x = gmpy2.mpz(x)
     ...:   return gmpy2.bit_length(x&-x)-1

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

In [3385]: def checkifperfectnum(N):
     ...:    a = ffs(N)
     ...:    e = extractoddfactor(N)
     ...:    ex = 2**(a+1)-1
     ...:    if e == ex: return True
     ...:    else: return False

In [3383]: 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))
     ...: 
     ...:

Answer:

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

you'll see the 15, 63, 1023, etc do not work with the odd cube method.

So every second odd number n in the form of (2**(n-1)*(2**n-1) is true for this equation (making 6 being the exception). This would obviously include every even perfect number that is a Mersenne prime. While stating every Centered nonagonal number is true, this could be expanded to the exact statement statement at the beginning of this paragraph.

So I think what I'm saying is that there is an expanded, more accurate statement to be made of the odd cube method, that doesn't require it to be tested if it's a centered nonagonal number since every odd n is a centered nonagonal number. This can be verified via:

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

The Centered nonagonal number wiki doesn't mention that it includes every odd n in the form of (2**(n-1)*(2**n-1) either so I don't think anyone would come to that determination without doing the math. I'm not sure why it's not mentioned, unless there is no published proof of it, maybe?

LeagueEnthusiast (talk) 05:08, 6 May 2022 (UTC)LeagueEnthusiast[reply]

Another method of deriving even perfect numbers[edit]

The following equation in this program will derive all even perfect numbers using 2**number-1:

In [3394]: 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 [3396]: for x in range(2,16):
     ...:     print (altpnusewithnumbertopower(x))
     ...: 
6
28
120
496
2016
8128
32640
130816
523776
2096128
8386560
33550336
134209536
536854528

LeagueEnthusiast (talk) 05:11, 6 May 2022 (UTC)LeagueEnthusiast[reply]

There are no odd perfect numbers.[edit]

There are no such numbers![1]https://arxiv.org/abs/2101.07176 I am a Green Bee (talk) 10:09, 24 July 2023 (UTC)[reply]

@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)[reply]

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)[reply]

New paper by Clayton and Hansen[edit]

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)[reply]

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

Ditto, go for it. --JBL (talk) 19:04, 27 November 2023 (UTC)[reply]

rename[edit]

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)[reply]

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)[reply]