IMPROVED TECHNIQUES FOR LOWER BOUNDS FOR ODD PERFECT NUMBERS

RICHARD P. BRENT, GRAEME L. COHEN, AND H. J. J. TE RIELE

Abstract

If N is an odd perfect number, and q^{k}∥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 [1] to show that N > 10^{300}.

Comments

Only the Abstract is given here. The full paper appeared as [2]. 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 12655 leaves gives either a contradiction or a sufficiently large lower bound on N. The proof tree is available by anonymous ftp in a separate file rpb116p.dvi.Z . The integer factorizations used in the proof are also available by anonymous ftp: see [3].

References

1. R. P. Brent, and G. L. Cohen, “A new lower bound for odd perfect numbers”, Mathematics of Computation 53 (1989), 431–437. Supplement, ibid, S7–S24. MR 89m:11008. Also appeared as Report TR-CS-88-05, Computer Sciences Laboratory, ANU, February 1988, 50 pp. rpb100.

2. 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. Also appeared as Report NM-R8921, Centrum voor Wiskunde en Informatica, Amsterdam, October 1989, 13 pp. A longer version was submitted to the Mathematics of Computation UMT file and appeared as Report CMA-R50-89, CMA, ANU, October 1989, 198 pp. rpb116.

3. R. P. Brent, Factor: An Integer Factorization Program for the IBM PC, Report TR-CS-89-23, Computer Sciences Laboratory, ANU, October 1989, 7 pp. rpb117.

(Brent) Computer Sciences Laboratory, Australian National University, Canberra E-mail address: rpb@cslab.anu.edu.au

(Cohen) School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia

(te Riele) Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands

E-mail address: herman@cwi.nl

1991 Mathematics Subject Classification. Primary 11A25; Secondary 11-04, 11Y05, 11Y70. Key words and phrases. Perfect numbers, odd perfect numbers, lower bounds, integer factorization,

proof tree.

Some of the factorizations required for the proof were obtained using the Fujitsu VP-100 of the Australian National University Supercomputer Facility.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1991, American Mathematical Society. Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb116a typeset using AMS-L^{A}TEX