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An Extension of the Euclid-Euler Theorem to
Certain $\alpha$-Perfect Numbers Paulo J. Almeida¹ and Gabriel Cardoso²
University of Aveiro
Department of Mathematics
Campus Universitário de Santiago
3810-193 Aveiro
Portugal
palmeida@ua.pt
gabriel.cardoso@ua.pt

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Abstract:

In a posthumously published work, Euler proved that all even perfect numbers are of the form $2^{p-1}(2^p-1)$, where $2^p-1$ is a prime number. In this article, we extend Euler's method for certain $\alpha$-perfect numbers for which Euler's result can be generalized. In particular, we use Euler's method to prove that if $N$ is a $3$-perfect number divisible by $6$; then either $2\parallel N$ or $3\parallel N$. As well, we prove that if $N$ is a $\frac{5}{2}$-perfect number divisible by $5$, then $2^4\parallel N$, $5^2\parallel N$, and $31^2\mid N$. Finally, for $p\in\{17,257,65537\}$, we prove that there are no $\frac{2p}{p-1}$-perfect numbers divisible by $p$.