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

MathSciNet review:
1094940

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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]**Richard P. Brent and Graeme L. Cohen,*A new lower bound for odd perfect numbers*, Math. Comp.**53**(1989), no. 187, 431–437, S7–S24. MR**968150**, 10.1090/S0025-5718-1989-0968150-2**[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]**John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr.,*Factorizations of 𝑏ⁿ±1*, 2nd ed., Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, RI, 1988. 𝑏=2,3,5,6,7,10,11,12 up to high powers. MR**996414****[6]**Peter Hagis Jr.,*Outline of a proof that every odd perfect number has at least eight prime factors*, Math. Comp.**35**(1980), no. 151, 1027–1032. MR**572873**, 10.1090/S0025-5718-1980-0572873-9**[7]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 4th ed., Oxford Univ. Press, 1962.

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

Article copyright:
© Copyright 1991
American Mathematical Society