A new lower bound for odd perfect numbers

Abstract

We describe an algorithm for proving that there is no odd perfect number less than a given bound <italic>K</italic> (or finding such a number if one exists). A program implementing the algorithm has been run successfully with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K equals 10 Superscript 160"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>160</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K = {10^{160}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with an elliptic curve method used for the vast number of factorizations required.