An Extension of the Euclid-Euler Theorem to Certain α-Perfect Numbers

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

-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 $p$

An Extension of the Euclid-Euler Theorem to Certain α-Perfect Numbers

Paulo J. Almeida and Gabriel Cardoso University of Aveiro Department of Mathematics Campus Universitário de Santiago 3810-193 Aveiro Portugal

Paulo J. Almeida and Gabriel Cardoso
University of Aveiro
Department of Mathematics
Campus Universitário de Santiago
3810-193 Aveiro
Portugal