Wikipedia:Reference desk/Archives/Mathematics/2024 October 21

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October 21

52nd perfect number

How many digits (I want an exact figure) does the 52nd perfect number have?? Georgia guy (talk) 13:11, 21 October 2024 (UTC)

If you read the perfect number article you will see that only 51 perfect numbers are known. So nobody knows. 196.50.199.218 (talk) 13:38, 21 October 2024 (UTC)

Please, I learned this morning that a new perfect number has been discovered. Georgia guy (talk) 13:41, 21 October 2024 (UTC)

Although a possible 52nd Mersenne prime has been discovered, its primality has not been ascertained and its identity has not been released, so we cannot construct a perfect number from it yet. Also, after the 48th Mersenne prime, we get into unverified territory, meaning that there may be additional Mersenne numbers between the Mersenne primes we know about that are also prime, but that we missed. GalacticShoe (talk) 13:42, 21 October 2024 (UTC)

It was revealed this morning to be prime. Georgia guy (talk) 13:44, 21 October 2024 (UTC)

Well, do you have the value of $n$ that they found produces the new prime $2^{n}-1$? If so then the number of digits is going to be $\lfloor \log _{10}(2^{n-1}(2^{n}-1))\rfloor +1=\lfloor (n-1)\log _{10}(2)+\log _{10}(2^{n}-1)\rfloor +1\approx \lceil (2n-1)\log _{10}(2)\rceil$. GalacticShoe (talk) 13:53, 21 October 2024 (UTC)

GalacticShoe, I don't want a formula; I want an answer; I believe it's more than 80 million but I want an exact figure. Georgia guy (talk) 13:55, 21 October 2024 (UTC)

I see someone has updated the Mersenne prime page with the value $n=136279841$. If you plug that into the formula I provided, you get $82048640$ digits. GalacticShoe (talk) 14:00, 21 October 2024 (UTC)

@GalacticShoe: I added your figure to List of Mersenne primes and perfect numbers. Still need the digits of the perfect number, though. :) Double sharp (talk) 14:29, 21 October 2024 (UTC)

Thanks, Double sharp. Unfortunately, I don't think my computer could handle that kind of number so I'll have to deign to someone else for this one :) GalacticShoe (talk) 14:41, 21 October 2024 (UTC)

Well, we only need the first six and last six digits for consistency in the table. Wolfram Alpha is giving me 388692 for the first six digits, and it must end in ...008576 by computing modulo 10⁶.

And now I realise that the GIMPS press release links to a zip file containing the perfect number as well. Oops. Well, nice to know for sure that the above is correct. Double sharp (talk) 14:52, 21 October 2024 (UTC)

Now that I think further, it's actually pretty simple to find the first 6 digits, since all you have to do is take $136279841$, plug it into $a=(2n-1)\log _{10}(2)$ to get the approximate base-10 exponent of the perfect number, then find the first six digits of $10^{a-b}$ where $b$ is an integer offset that allows us to scale the perfect number down by an arbitrary power of 10. Doing so with $b=82048634$ yields the aforementioned $388692$. GalacticShoe (talk) 15:08, 21 October 2024 (UTC)

Using home-brewed routines, I get 3886924435...2330008576. I can produce some more digits if desired, up to several hundreds, but not all 82048640 of them.  --Lambiam 17:01, 21 October 2024 (UTC)

The first 200 digits are

38869244357621839661971659704624268855322994779882

08304859499151240125532614860002576774959569836297

23543288883178333212483499264905902862070554869707

09549201405173953201448117706188781295031055831985

The last 200 digits are

26371148194474346970940249611191311736905503524083

34567573632858973741715968532307863945069401854756

19103726629021654022624182624160188048957025553161

11949805495520297305681205442638972750732330008576

Use PARI/GP programs:

First 200 digits:

localprec(250);10^frac(272559681*log(2)/log(10))\10^-199

Last 200 digits:

lift(Mod(2,10^200)^136279840*(Mod(2,10^200)^136279841-1))

-- 220.132.216.52 (talk) 10:00, 2 November 2024 (UTC)

@Lambiam: Your 10th last digit is wrong, it is 2 instead of 7. 220.132.216.52 (talk) 15:57, 4 November 2024 (UTC)

Thanks, now corrected.  --Lambiam 19:49, 4 November 2024 (UTC)

the zip file for the perfect number 220.132.216.52 (talk) 09:12, 2 November 2024 (UTC)