Improved techniques for lower bounds for odd perfect numbers

Authors:
R. P. Brent, G. L. Cohen and H. J. J. te Riele

Journal:
Math. Comp. **57** (1991), 857-868

MSC:
Primary 11A25; Secondary 11Y70

DOI:
https://doi.org/10.1090/S0025-5718-1991-1094940-3

MathSciNet review:
1094940

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**[1]**R. P. Brent,*Factor*:*an integer factorization program for the*IBM PC, Report TR-CS-89-23, Computer Sciences Laboratory, Australian National University, October 1989.**[2]**R. P. Brent and G. L. Cohen,*A new lower bound for odd perfect numbers*, Math. Comp.**53**(1989), 431-437, S7-S24. MR**968150 (89m:11008)****[3]**R. P. Brent, G. L. Cohen, and H. J. J. te Riele,*An improved technique for lower bounds for odd perfect numbers*, Report TR-CS-88-08, Computer Sciences Laboratory, Australian National University, August 1988.**[4]**R. P. Brent, G. L. Cohen, and H. J. J. te Riele,*Improved techniques for lower bounds for odd perfect numbers*, Report CMA-R50-89, Centre for Mathematical Analysis, Australian National University, October 1989.**[5]**J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr.,*Factorizations of**up to high powers*, 2nd ed., Contemp. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1988. MR**996414 (90d:11009)****[6]**P. Hagis, Jr.,*Outline of a proof that every odd perfect number has at least eight prime factors*, Math. Comp.**34**(1980), 1027-1032. MR**572873 (81k:10004)****[7]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 4th ed., Oxford Univ. Press, 1962.

Retrieve articles in *Mathematics of Computation*
with MSC:
11A25,
11Y70

Retrieve articles in all journals with MSC: 11A25, 11Y70

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1991-1094940-3

Article copyright:
© Copyright 1991
American Mathematical Society