Journal of Integer Sequences, Vol. 25 (2022), Article 22.8.4

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


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 $p$.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A005820 A019434 A214409 A214413.)

Received July 2 2022; revised version received July 3 2022; October 5 2022; October 6 2022; October 12 2022. Published in Journal of Integer Sequences, October 14 2022.

Return to Journal of Integer Sequences home page