Improved techniques for lower bounds for odd perfect numbers

116. R. P. Brent, G. L. Cohen and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Mathematics of Computation 57 (1991), 857-868. MR 92c:11004.

Longer version (including the proof tree) appeared as Report CMA-R50-89, Centre for Mathematical Analysis, ANU, October 1989, 198 pp.

Abstract: dvi (3K), pdf (89K), ps (31K).

Paper: dvi (20K), pdf (356K), ps (69K).

Appendix (the proof tree, 194 pp.): dvi (190K), pdf (812K), ps (565K), gzipped tex (170K),


If N is an odd perfect number, and qk is the highest power of q dividing N, where q is prime and k is even, then it is almost immediate that N > q2k. We prove here that, subject to certain conditions verifiable in polynomial time, in fact N > q5k/2. Using this and related results, we are able to extend the computations in an earlier paper [100] to show that N > 10300.


The main part of the proof that there is no odd perfect number N less than 10300 is a (very large) tree, each of whose leaves gives either a contradiction or a sufficiently large lower bound on N. The proof tree is available as a separate file (see above). The number of leaves has been reduced from 12655 (as stated in the paper) to 12644 by using some additional factorizations.

The integer factorizations used in the proof are available separately.

This paper improved several earlier results [100, 106].

Added January 2011: Pascal Ochem and Michael Rao have considerably improved the result, aiming to show that N > 101500. Details are available here.

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