Leonhard Euler, after proving that every even perfect number has the form given by Euclid, turned his attention to finding odd perfect numbers. Euler established a basic factorization pattern that every odd perfect number must have, and mathematicians have expanded upon this Eulerian form ever since. This paper will present a brief summary of Euler’s result and some recent generalizations. It will also note connections between odd perfect numbers and the abundancy index (the abundancy index of a positive integer is the ratio of the sum of its positive divisors to itself). In particular, finding a positive integer with an abundancy index of 5/3 would finally produce that elusive odd perfect number.
Beasley, Brian D., "Euler and the Ongoing Search for Odd Perfect Numbers" (2013). ACMS Conference Proceedings 2013. 11.
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