15 research outputs found
On the total number of prime factors of an odd perfect number
We say n ∈ ℕ is perfect if σ (n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n = pα Πkj=1 q2βjj , where p, q1, ⋯, qk, are distinct primes and p ≡ α ≡ 1 (mod 4). We prove that if βj ≡ 1 (mod 3) or βj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3 ∦ n. We also prove as our main result that Ω(n) ≥ 37, where Ω(n) = α + 2∑kj=1 βj. This improves a result of Sayers (Ω(n) ≥ 29) given in 1986
Prime factors of of the same form
We parameterize solutions to the equality
when each is
prime. Our focus is on the special cases when , as this analysis
simplifies and extends bounds on the total number of prime factors of an odd
perfect number
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
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
On the small prime factors of a non-deficient number
Let to be the sum of the positive divisors of . A number is
non-deficient if . We establish new lower bounds for the
number of distinct prime factors of an odd non-deficient number in terms of its
second smallest, third smallest and fourth smallest prime factors. We also
obtain tighter bounds for odd perfect numbers. We also discuss the behavior of
, , and related sequences.Comment: 18 page