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Old 2021-11-14, 23:00   #1
jonne
 
Nov 2021

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Question Logarithm of logarithm of perfect numbers?

Hello everyone, this is my first post.

I saw a graph with logarithms of logarithms of perfect numbers and they followed a line defined by some logarithm(s) and the Euler-Mascheroni constant. Unfortunately I don't remember anything else from that paper so I can't find it.

I want to create same kind of a graph for my thesis about perfect numbers but computing the logarithms of logarithms of the larger perfect numbers is too much for my laptop, result was memory error when trying to compute some with python.

So my question is, if the logs of logs are available somewhere or if it is possible to do such computations with the GIMPS-related software. I don't have to pay for electricity in my student flat so if GIMPS software can do this kind of computing slowly, it is no problem.

I would also appreciate someone pointing out the paper or some other paper(s) that feature the log-log graph of perfect numbers.

I thought this 'Math' subsection would be the best place for this question. Feel free to move it, if there's some better place for it. Computational number theory also passed my mind as a proper place for posting this.

Moderator note: Moved to Homework Help

Last fiddled with by Dr Sardonicus on 2021-11-14 at 23:59
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Old 2021-11-14, 23:59   #2
retina
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All known perfect numbers can be derived from Mersenne primes, (2^p-1)*(2^(p-1))

In base 2 the log is approximately 2p-1, so for all known MP the magnitude of the first logs are no larger than 9 digits. No need for anything more than a 9 digit calculator.
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Old 2021-11-15, 14:57   #3
R. Gerbicz
 
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You want this page from Iron Age: https://primes.utm.edu/notes/faq/NextMersenne.html
for even(!) perfect numbers you get a similar graph, since P(k)=2^(p-1)*(2^p-1)~0.5*Mp^2
and for that log2(log2(P(k)) is very close to 1+log2(log2(Mp)), you get a parallel line.
Up to a pretty large bound there is no odd perfect number, probably there is none, so these graphs are the same.
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