A new lower bound for odd perfect numbers

100. R. P. Brent and G. L. Cohen, A new lower bound for odd perfect numbers, Mathematics of Computation 53 (1989), 431-437. Microfiche supplement ibid, S7-24. MR 89m:11008.

Abstract: dvi (3K), pdf (79K), ps (28K).

Paper: pdf (724K).

Supplement (the proof tree): pdf (2037K).


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 = 10160, with the elliptic curve method used for the vast number of factorizations required.


The proof tree has 3132 leaves and is available as a separate file (see above).

For a sequel which extended the result to K = 10300, see Brent, Cohen and te Riele [116]. The integer factorizations used in the proofs are available here.

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