A NEW LOWER BOUND FOR ODD PERFECT NUMBERS

RICHARD P. BRENT AND GRAEME L. COHEN

Abstract

We describe an algorithm for proving that there is no odd perfect number less than a given bound K (or finding such a number if one exists). A program implementing the algorithm has been run successfully with K = 10^{160}, with an elliptic curve method used for the vast number of factorizations required.

Comments

Only the Abstract is given here. The full paper appeared as [1]. For a sequel which extended the result to K = 10^{300}, see [2]. The integer factorizations used in the proofs are available by anonymous ftp [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. 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, Australia

E-mail address: rpb@cslab.anu.edu.au

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

E-mail address: glcohen@utscsd.oz

1991 Mathematics Subject Classification. Primary 11-04; Secondary 11A25, 11A51, 11Y05, 11Y70. Key words and phrases. Perfect number, odd perfect number, lower bound, tree search, integer fac-

torization, elliptic curve method, ECM.

The research was carried out while the second author was a Visiting Fellow at the Australian National University.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1989, American Mathematical Society. Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb100a typeset using AMS-L^{A}TEX