Abstract by Jonathan Hales
BYU Computational Number Theory Group BYU Computational Number Theory Group
Spoof Odd Perfect Numbers
In 2014 Sam Dittmer introduced a generalization to the odd perfect number problem known as spoof perfect numbers. This notion is motivated by a 1638 letter from Descartes to Mersenne in which he notices that the number D = 198585576189 = 3^2 7^2 11^2 13^2 22021 would be an odd perfect number if one mistakingly thought that the number 22021 = 19^2 61 were prime. We make this idea precise and give a complete list of all odd spoof perfect numbers with less than seven bases. We show that the structure of odd spoofs is extremely rich and is composed of multiple infinite families. This implies that many approaches to the odd perfect number problem are intractable. In particular any attempt to solve the problem that uses only the multiplicative nature of the sum-of-divisors function will have infinitely many counter-examples.