## 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),

## Abstract

If *N* is an odd perfect number, and
*q*^{k} is the highest power of
*q* dividing *N*,
where *q* is prime and *k* is 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
[100]
to show that *N* > 10^{300}.
## Comments

The main part of the proof that there is no odd perfect number *N*
less than 10^{300} 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* > 10^{1500}.
Details are available
here.

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