66,402 research outputs found
Odd Perfect Numbers Have At Least Nine Distinct Prime Factors
An odd perfect number, N, is shown to have at least nine distinct prime
factors. If 3 does not divide N, then N must have at least twelve distinct
prime divisors. The proof ultimately avoids previous computational results for
odd perfect numbers.Comment: 17 page
Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers
In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980.
What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that 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
Primitive abundant and weird numbers with many prime factors
We give an algorithm to enumerate all primitive abundant numbers (briefly,
PANs) with a fixed (the number of prime factors counted with their
multiplicity), and explicitly find all PANs up to , count all PANs
and square-free PANs up to and count all odd PANs and odd
square-free PANs up to . We find primitive weird numbers (briefly,
PWNs) with up to 16 prime factors, improving the previous results of
[Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been
given. The largest PWN we find has 14712 digits: as far as we know, this is the
largest example existing, the previous one being 5328 digits long [Melfi]. We
find hundreds of PWNs with exactly one square odd prime factor: as far as we
know, only five were known before. We find all PWNs with at least one odd prime
factor with multiplicity greater than one and and prove that there
are none with . Regarding PWNs with a cubic (or higher) odd prime
factor, we prove that there are none with , and we did not find
any with larger . Finally, we find several PWNs with 2 square odd prime
factors, and one with 3 square odd prime factors. These are the first such
examples.Comment: New section on open problems. A mistake in table 2 corrected (# odd
PAN with Omega=8). New PWN in table 5, last line, 2 squared prime factors,
Omega=15. Updated bibliograph
Must a primitive non-deficient number have a component not much larger than its radical?
Let be a primitive non-deficient number, with where the are distinct primes. Let We prove that there must be an such that . We
conjecture that there is always an such that and prove
this stronger inequality in some cases.Comment: 10 page
- …