Improved Techniques for Lower Bounds for Odd Perfect Numbers

Abstract

Abstract. If N is an odd perfect number, and q \\ N, q prime, k even, 2k then it is almost immediate that N > q .We prove here that, subject to certain conditions verifiable in polynomial time, in fact N > q ' . Using this and related results, we are able to extend the computations in an earlier paper to show that N > 10300 .

Odd harmonic numbers exceed 10²⁴

“…The bound is currently 10 300 . See Brent and Cohen [1], and Brent, Cohen and te Riele [2]. In the latter paper, a conditional improvement on the result n > p 2a is discussed and used.…”

Factorization of the tenth Fermat number

“…Operations which are not time-critical, such as input and output, are performed using the MP package [5]. Program C found the factorization of F 11 (see §7) and many factors, of size up to 40 decimal digits, needed for [16,19]. Keller [37] used program C to find factors up to 39 digits of Cullen numbers.…”

“…For example, 27-digit factors of F 13 and F 16 have recently been found [13,17]. The smallest Fermat number which is not yet completely factored is F 12 .…”

A Class of New Near-Perfect Numbers

“…Thus from (3.3) we know that (2 α+1 − 1)(p 2 1 + p 1 + 1) = 2 α+1 p 2 1 − k + 2 β p γ 1 , namely, 2 β p γ 1 = (2 α+1 − 1)(p 1 + 1) + k − p 2 1 . Thus we complete the proof of (3).…”

“…For odd perfect numbers, Euler obtained a necessary condition for the existence (see [5]). In recent years, there have been many papers for odd perfect numbers having to do with the conjecture that there exists no odd perfect numbers (see [2,3,4,7,12]). Until now, the conjecture has not been proved.…”