Abstract. If N is an odd perfect number, and q \\ N, q prime, k even, 2k then it is almost immediate that N > q .We prove here that, subject to certain conditions verifiable in polynomial time, in fact N > q ' . Using this and related results, we are able to extend the computations in an earlier paper to show that N > 10300 .
Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 .Jesmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an) x + (bn) v = (en)* in positive integers is x -y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > l , c = b + l and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples
New methods are introduced here to show that if n is a quasiperfect number and u(n) the number of its distinct prime factors, then u(n) > 7 and n > 10 35 , and if further 3}« then u(n) > 9 and n > 1O 40 .
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 = 10160, with an elliptic curve method used for the vast number of factorizations required.
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