Talk:Perfect number

Gallardo's Result
Is Gallardo's result (in Minor results) true? In the linked paper he implicitly assumed that $$x + a$$ and $$x^2 - a x + a^2$$ 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)


 * 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)

Changes to odd perfect number section and COI
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)


 * 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)
 * 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)

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 $$N $$ 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)


 * 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 $$a < N^\frac15.$$ 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 $$a \ge 101.$$ 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)


 * 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)


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

conjecture regarding the divisors of Perfect Numbers (PN)
I conjecture that the product of the divisors a PN derived from 2p-1(2p - 1) will equal PNp-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)
 * 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)
 * Note that the term divisor includes the number itself so their product becomes PNp. 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 2m or 2m×(2p - 1). The divisors without the number itself are called the proper divisors.
 * Regarding your other observation, it's known that 2p-1(2p - 1) is a perfect number if and only if 2p - 1 is prime (called a Mersenne prime). The only odd divisors of 2p-1(2p - 1) are 1 and 2p - 1. Your observation follows from this so it persists. PrimeHunter (talk) 04:47, 26 December 2021 (UTC)

I think the Odd Cubes section could be made more accurate
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

Another method of deriving even perfect numbers
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

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)
 * 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 = 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)