Odd perfect numbers, Diophantine equations, and upper bounds

Nielsen, Pace

doi:10.1090/S0025-5718-2015-02941-X

Odd perfect numbers, Diophantine equations, and upper bounds
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Math. Comp. 84 (2015), 2549-2567 Request permission

Abstract:

We obtain a new upper bound for odd multiperfect numbers. If $N$ is an odd perfect number with $k$ distinct prime divisors and $P$ is its largest prime divisor, we find as a corollary that $10^{12}P^{2}N<2^{4^{k}}$. Using this new bound, and extensive computations, we derive the inequality $k\geq 10$.

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