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March 12

Odd perfect numbers

For some reason, people are continuing to accept the idea that there are odd perfect numbers and that we just haven't found any of them. Go to oddperfect.org and see if you can answer the question "Are they depending on finding an OPN or a proof that there are no OPN's??" Georgia guy (talk) 15:53, 12 March 2014 (UTC)[reply]

They explain their objection to the heuristic argument of Carl Pomerance here. Count Iblis (talk) 16:30, 12 March 2014 (UTC)[reply]

And it's not really a strong objection:

"The Pomerance Heuristic suggests there should be no large perfect numbers — neither even nore odd. Dr. Pomerance argues that Mersenne Primes are a grand conspiracy to circumvent the heuristic's randomness assumption, and that no such conspiracy exists for odd perfect numbers.

Personally, I find this argument strong enough that I intend to bias the odd perfect search in directions that will produce a higher incidence of useful side effects in the form of collectible factorizations. But I don't find the argument sufficiently overwhelming to cancel the search entirely. Advances in factoring technology have created new "low hanging fruit" that we can harvest and hope to uncover an odd conspiracy."

Count Iblis (talk) 16:32, 12 March 2014 (UTC)[reply]

Wikipedia does not know of any odd perfect numbers, but nor do we have any proof that such numbers do not exist. The question of whether there are any odd perfect numbers is a well-known unsolved problem. Your question was "Are they depending on finding an OPN or a proof that there are no OPN's??". The answer is that mathematicians rarely "depend" on anything. Rather, they are interested in learning whether particular results can be proven true or false. Because of the way mathematics works, even a negative result (proving that there are no odd perfect numbers) would probably still be worthwhile, because proving this would have required the development of tools and techniques (for example in factorization theory) that could be applied in other areas. You will notice that the website you mentioned (www.oddperfect.org) deliberately aims to produce tools and results that will be useful even if their main search fails. RomanSpa (talk) 14:01, 13 March 2014 (UTC)[reply]

There is a view that mathematicians are interested only in what can be proved or disproved. That view is, to use the technical term, "wrong". Mathematicians are also interested in what is true or false, even when proofs are not available, and arguments such as the one Iblis alludes to are part of that discussion. --Trovatore (talk) 20:38, 13 March 2014 (UTC)[reply]

The alert reader will note that I did not say that mathematicians are only interested in what can be proved or disproved. There is a substantial debate about what mathematics is and what mathematicians are interested in; a good place to start is with our article on the Philosophy of Mathematics. Certainly, heuristic arguments are useful in mathematics, particularly when guiding a student towards research choices - I'd strongly discourage any student from dedicating his PhD years specifically to a brute-force search for an OPN, for example - but by their nature such arguments do not end up in the book. As it happens, in the first draft of my response to this question I wrote a paragraph on Carl Pomerance's argument, before realising that discussion of it did not answer the original question: it is fairly clear that the website mentioned in the original question is interested in proof, specifically by identifying an OPN, and mentions Pomerance's heuristic only to set it to one side. RomanSpa (talk) 23:33, 13 March 2014 (UTC)[reply]