Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. $\sigma_1 (n) = 2n$. So one can easily define an algorithm to check for perfect numbers :

May 3, 2023 · As far as I can judge, when a number is perfect, then the sum of its divisors is the double of that number, so it can't have any other new prime factors (except for $2$, obviously).

I came up with these: Applications of Mersenne numbers: signed/unsigned integers, towers of Hanoi Applications of Fermat numbers: relation to constructible polygons But for perfect numbers the best I …

A well known theorem by Euler states that every even perfect number is of the form $2^ {p-1} (2^p-1)$ where $2^p-1$ is prime (this is what is called a Mersenne prime).

Dec 5, 2019 · There has been a lot of work done on this topic, enough so it is hard to summarize in a forum like this. If you search for odd perfect number you will find a lot of information. Some of it is …

Feb 10, 2016 · I don't think there are any perfect prime numbers. And The question specifically says n is and even perfect number.

Nov 22, 2024 · The only prime divisors of an (even) perfect number are $2$ and the Mersenne prime associated with the perfect number. What are you asking?

@user129901 A proof of Euler's result that all even perfect numbers are in fact of the form (the form that Euclid knew), can be found online several places, for example A proof that all even perfect numbers …

Oct 16, 2018 · A perfect number is a number than can be expressed as a sum of its factors. For example, 28 = 1 + 2 + 4 + 7 + 14 Why is this property important? What is so perfect about perfect …

The question Are there infinitely many perfect numbers? is a classic old unsolved problem. However, we keep finding perfect numbers (via Mersenne primes) and produce a lot of knowledge on perfect n...