Improved techniques for lower bounds for odd perfect numbers
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If N is an odd perfect number, and qk‖N , q prime, k 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 [1] to show that N > 10300. 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 10300 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 Nether- lands E-mail address: herman@cwi.nl 1991 Mathematics Subject Classification. Primary 11A25; Secondary 11-04, 11Y05, 11Y70.
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