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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
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Abstract: If N is an odd perfect number, and $ {q^k}\vert\vert N$, q prime, k even, then it is almost immediate that $ N > {q^{2k}}$. We prove here that, subject to certain conditions verifiable in polynomial time, in fact $ N > {q^{5k/2}}$. Using this and related results, we are able to extend the computations in an earlier paper to show that $ N > {10^{300}}$.


References [Enhancements On Off] (What's this?)

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  • [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.
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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1094940-3
Article copyright: © Copyright 1991 American Mathematical Society

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