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January 19

49th perfect number

How many digits does the 49th perfect number have?? I say more than 40 million, am I right?? Georgia guy (talk) 15:56, 19 January 2016 (UTC)[reply]

From Perfect number#Even perfect numbers:

Four higher perfect numbers have also been discovered, namely those for which p = 37156667, 42643801, 43112609, and 57885161, though there may be others within this range.

and

48 even perfect numbers (the largest of which is $2^{57885160}\times (2^{57885161}-1)$ with 34,850,340 digits).

Thus for all we know the 49th one could be the one that is currently the largest known, with less than 35 million digits, or even smaller than that. Or maybe not. Loraof (talk) 16:51, 19 January 2016 (UTC)[reply]

That's the 48th, not the 49th. The 49th was just discovered, and it is larger. Georgia guy (talk) 16:55, 19 January 2016 (UTC)[reply]

Loraof appears to be making the point that the list of known perfect numbers is not believed to be complete, which means there is a good chance that the 49th smallest perfect number is actually smaller than the largest perfect number that is currently known. Dragons flight (talk) 20:58, 19 January 2016 (UTC)[reply]

According to List of perfect numbers (recently updated by PrimeHunter), it has 44,677,235 digits. -- Meni Rosenfeld (talk) 20:49, 19 January 2016 (UTC)[reply]

Yes, after today's announcement [1] the 49th known perfect number is 2^74,207,280 × (2^74,207,281 − 1), and (74,207,280 + 74,207,281) × log(2)/log(10) = 44677234.65430... PrimeHunter (talk) 21:12, 19 January 2016 (UTC)[reply]

As long as we're here, can you explain the gap between the announcement date (19) and finding date (7)? Was there a review process in this time, during which the finding is kept secret? -- Meni Rosenfeld (talk) 23:30, 19 January 2016 (UTC)[reply]

There are checks with independent software and hardware before the number is published. The linked announcement says the checks took 3-4 days so I don't know why it was published 12 days later. 9 January it was posted in a public project forum that a prime had been reported.[2] There is probably more about the process there. PrimeHunter (talk) 00:48, 20 January 2016 (UTC)[reply]

Hm, seems the situation is even worse - "The prime was reported last year but went unnoticed (again) for months". -- Meni Rosenfeld (talk) 10:01, 20 January 2016 (UTC)[reply]

In general, a perfect number has twice as many digits as the associated Mersenne prime, or one fewer than twice as many. Bubba73 ^{You talkin' to me?} 19:38, 20 January 2016 (UTC)[reply]

Or two fewer. If the Mersenne prime is q then the perfect number is approximately q²/2. If the leading digits of q are less than those of sqrt(2) = 1.41421..., then the perfect number gets two fewer digits. Benford's law means it will probably happen quite frequently. Looking at Mersenne prime#List of known Mersenne primes it happens for rank 4, 6, 15, 33, 37, 40, 42, 44. Compare to List of perfect numbers. Rank 4 is the Mersenne prime 127 with perfect number 8128. PrimeHunter (talk) 10:43, 21 January 2016 (UTC)[reply]