OddPerfect.org
The Arguments Against

# I. Urban Legends

It is an urban legend that there is overwhelming evidence that odd perfect numbers do not exist. At least I think it's an urban legend because I haven't been able to find the overwhelming evidence. If you know of overwhelming evidence not discussed here, please email William at Odd Perfect dot Org.

I'm aware of two arguments that odd perfect do not exist: Sylvester's "web of conditions" and Pomerance's Heuristic. Sylvester's web is discussed on this page, while Pomerance's Heuristic is discussed on the Pomerance page. Sylvester's web, at least as usually described, implies a constant difficulty that will eventually fall to higher number searches. Pomerance's heuristic indicates there should be no large perfect numbers — neither even nor odd. Pomerance makes arguments for why the heuristic could be wrong for even perfect numbers but right for odd perfect numbers. Although I can understand some people finding the Pomerance Heuristic a "strong argument," I contend that such a conveniently nuanced belief cannot be judged "overwhelming evidence."

# II. Sylvester's Web

According to Gimbel and Jaroma, "much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888." James Joseph Sylvester is famous for his eloquence, and his 1888 statement on the unlikeliness of odd perfect numbers is often quoted:

"… a prolonged meditation on the subject has satsfied me that the existence
of any one such — its escape, so to say, from the complex web of conditions
which hem it in on all sides — would be little short of a miracle."

This quote is usually followed by a modern list of the conditions which hem it in on all sides. For example:

1. N = p4λ+1 Q2 (Euler)
2. At least 9 distinct prime factors (Nielsen)
3. At least 75 prime factors, counting multiplicities. (Hare)
4. Greater than 10300 (Brent, Cohen, the te Riele)
5. Less than 24k, k distinct primes (Nielsen)
6. Largest prime exceeds 108 (Goto & Ohno)
7. Second largest prime exceeds 10,000 (Iannucci)
8. Third largest prime exceeds 100 (Iannucci)
9. Largest component exceeds 1020 (Cohen)
10. Largest exponent at least 4 (Kanold)

# Is There Any Hope?

There is great hope because nothing on the list that scales badly with the size of N. Given only these constraints, if the probability a random 500 digit number is an odd perfect number is "p", then the same probability applies to random 1000 or 1,000,000 digit numbers. It doesn't matter how unlikely the list is because more candidates will always overwhelm that unlikelyhood. If the probability is one in 101000, then there is probably an odd perfect number with 1000 digits. If the list is so hard that it's one in 101,000,000, then there is probably a million digit odd perfect number.

The traditional roadblock to this difficulty is the need to factor thousand digit or million digit numbers. Our search innovation is to search the subset that has every component small enough to factor. This process can continue until the sheer number of known factors becomes too large to manage. It seems reasonable to hope that our search will overwhelm the list before the data management threshold is reached.

Although an eloquent testimonial to the difficulties of hand computing an odd perfect number, from the modern perspective eloquence is the only thing Sylvester's web has going for it. Personally, I prefer this quote (in translation) from Descartes:

"…and I don't know why you judge that this means the invention of
a true [odd] perfect number will not be successful … as for me, I
judge that one can find real odd perfect numbers."

## References

• R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Improved techniques for
lower bounds for odd perfect numbers, Math. Comp. 57 (1991), no. 196,
857–868. MR 1094940
• Cohen, G. L. "On the Largest Component of an Odd Perfect Number."
J. Austral. Math. Soc. Ser. A 42, 280-286, 1987.
• Takeshi Goto and Yasuo Ohno. Odd perfect numbers have a prime factor
exceeding \$10^8\$. Math. Comp. 77 (2008) 1859-1868.
• Kevin G. Hare. New techniques for bounds on the total number of prime
factors of an odd perfect number. Math. Comp. 76 (2007) 2241-2248.
• Iannucci, D. E. "The Second Largest Prime Divisor of an Odd Perfect Number
Exceeds Ten Thousand." Math. Comput. 68, 1749-1760, 1999.
• Iannucci, D. E. "The Third Largest Prime Divisor of an Odd Perfect Number
Exceeds One Hundred." Math. Comput. 69, 867-879, 2000.
• Kanold, H.-J. "Über mehrfach vollkommene Zahlen. II." J. reine angew. Math.
197, 82-96, 1957.
• Pace P. Nielsen. Odd perfect numbers have at least nine distinct prime
factors. Math. Comp. 76 (2007) 2109-2126.