A New Lower Bound for Odd Perfect Numbers

Abstract

Abstract.We describe an algorithm for proving that there is no odd perfect number less than a given bound K (or finding such a number if one exists). A program implementing the algorithm has been run successfully with K = 10160, with an elliptic curve method used for the vast number of factorizations required.

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.…”

“…A similar algorithm was used in [1] and [2]. Since the result is known to be true for p ≥ 37, the program runs through the possibilities p = 31, .…”

“…The 1 . The order of treatment of the first few primes is the order found to be convenient for odd perfect numbers in [1] and it is also similar to the approach adopted in [10].…”

Improved Techniques for Lower Bounds for Odd Perfect Numbers

“…The main result of this paper (Theorem 1 ) is still heavily dependent on the algorithm in [2], and we assume familiarity with that paper. It was stated at the end of that work that to continue the algorithm to obtain any substantial improvement of the earlier result required the factorization of the 81-decimal-72 digit composite number cr(13 ) ; this factorization has been completed and the result given in a postscript to [2].…”

Book Review: The book of prime number records

“…Many theorems give improbable properties of hypothetical odd perfect numbers. The record lower bound on the size of an odd perfect number is due to Brent and Cohen [1] who showed that any such number must exceed 10 150 . Exactly thirty-one perfect numbers are known, one for each Mersenne prime.…”