Talk:Perfect number

message: 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)
replies: text: 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)
thread_title:  Gallardo's Result 

message: 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)
thread_title:  I think the Odd Cubes section could be made more accurate 

message: 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)
thread_title:  Another method of deriving even perfect numbers 

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

message: 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)
replies: text: Properly published, so ok to use. I see no reason to object. —David Eppstein (talk) 19:01, 27 November 2023 (UTC)
replies: text: Ditto, go for it.  --JBL (talk) 19:04, 27 November 2023 (UTC)
thread_title:  New paper by Clayton and Hansen 

message: 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)
replies: text: 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)
thread_title:  rename 

message: 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)
replies: text: 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)
thread_title:  New perfect number found