Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [perfect-numbers]

Questions about or involving perfect numbers which are positive integers that are equal to the sum of their proper positive divisors.

A positive integer $n$ is said to be a perfect number if it is equal to the sum of its proper positive divisors.

The smallest example of a perfect number is $6$ as it has positive proper divisors $1$, $2$, $3$, and $1 + 2 + 3 = 6$.

More generally, $2^{p-1}(2^p-1)$ is perfect whenever $2^p - 1$ is a prime (called a Mersenne prime); the case above corresponds to $p = 2$. Furthermore, every even perfect number is of this form.

It is currently unknown whether there are infinitely many perfect numbers or whether any odd perfect numbers exist.

488 questions
20
votes
3 answers

When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

Let $\sigma(x)$ denote the sum of the divisors of $x$, and denote the abundancy index of $x$ as $$I(x) = \dfrac{\sigma(x)}{x},$$ and the deficiency of $x$ as $$D(x) = 2x - \sigma(x).$$ If the equation $I(a)=b/c$ has no solution $a \in \mathbb{N}$,…
Jose Arnaldo Bebita Dris
  • 8,471
20
votes
4 answers

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfect number (well it holds good for the first three perfect numbers…
user210387
17
votes
3 answers

If $N = q^k n^2$ is an odd perfect number and $n < q^{k+1}$, does it follow that $k > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, as of December 2018, there are $51$ known examples of even perfect numbers -- on the other hand, we still do not…
Jose Arnaldo Bebita Dris
  • 8,471
17
votes
4 answers

Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number of divisors which are odd. Adding them together…
James Pak
  • 1,547
13
votes
1 answer

Can an odd perfect number be divisible by $825$?

I know that an odd perfect number cannot be divisible by $105$. I wonder if that's also the case for $825$.
user77356
13
votes
3 answers

A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\sigma(z)$, and the sum of the aliquot divisors of…
Jose Arnaldo Bebita Dris
  • 8,471
12
votes
1 answer

How to show that all even perfect numbers are obtained via Mersenne primes?

A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). 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…
Gadi A
  • 19,839
11
votes
1 answer

Can an odd perfect number be divisible by $165$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
user77400
10
votes
1 answer

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to present my argument for this particular math…
Jose Arnaldo Bebita Dris
  • 8,471
10
votes
1 answer

How did Descartes come up with the spoof odd perfect number $198585576189$?

We call $n$ a spoof odd perfect number if $n$ is odd and and $n=km$ for two integers $k, m > 1$ such that $\sigma(k)(m + 1) = 2n$, where $\sigma$ is the sum-of-divisors function. In a letter to Mersenne dated November $15$, $1638$, Descartes showed…
Jose Arnaldo Bebita Dris
  • 8,471
10
votes
1 answer

On odd perfect numbers $n$ and $\sigma\left(n^\lambda\right)$

As background, I don't know if this kind of calculations were in the literature and/or are interestings. I can to prove that being $\lambda\geq 1$ a fixed integer, $n$ is perfect if and only if $$2n=\left(\prod_{p\mid…
user243301
9
votes
1 answer

Can an odd perfect number be divisible by $105$?

I have a tough one today. Show that if $n$ is an odd perfect number, then not all of $3$, $5$, and $7$ are divisors of $n$. Any and all help is appreciated. Thanks very much.
user39898
8
votes
2 answers

Discussion on even and odd perfect numbers.

First of all thank you so much for answering my previous post. These are few interesting problems drawn from Prof. Gandhi lecture notes. kindly discuss: 1) If $n$ is even perfect number then $(8n +1)$ is always a perfect square. 2) Every odd perfect…
Jiha
  • 107
7
votes
2 answers

Applications of Perfect Numbers

I'm preparing a talk on Mersenne primes, Perfect numbers and Fermat primes. In trying to provide motivation for it all I'd like to provide an application of these things. I came up with these: Applications of Mersenne numbers: signed/unsigned…
pdmclean
  • 809
7
votes
3 answers

Can powers of primes be perfect numbers?

I need to prove the following, though I'm not 100% certain I understand the definition of a perfect number. Prove that no perfect number is a power of a prime. First of all, I'm assuming that the question is asking me to prove that for any prime…
Mirrana
  • 9,299
1
2 3
…
32 33