# 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.

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### A Question Pertaining to Benjamin Peirce and Odd Perfect Numbers

It is known, but perhaps not too well known, that Benjamin Peirce demonstrated in 1832 that an odd perfect number must have at least four distinct prime divisors. In fact, he did so before a ...
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### Books/monographs/handouts specifically about perfect numbers and related topics

I am looking for books and/or monographs specifically about perfect numbers and related topics, where I could find proofs of results about them. An example that I have in mind for such a book is Song ...
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### If $N = q^k n^2$ is an odd perfect number with special prime $q$, then must $\sigma(q^k)$ be deficient?

The topic of odd perfect numbers likely needs no introduction. Let $\sigma=\sigma_{1}$ denote the classical sum of divisors. Denote the abundancy index by $I(x)=\sigma(x)/x$. An odd perfect number $N$...
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### What limit does the following inequality involving the sum of divisors and Euler totient functions reach?

Consider the following inequality: $$\frac{6}{{\pi}^2} < \frac{\sigma(n)\phi(n)}{n^2} \lt 1.$$ For the perfect numbers, this inequality seems to converge to 1. Here's the following ($5$ decimal) ...
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### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes: reference request or proposal as an exercise

I don't know if the following question is in the literature, please add a commment if it is in the literature (I add my thoughts and motivation below in last paragraph, it is discursive and ...
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### Is $n=6$ the only integer satisfying phenomenal properties in number theory ? if yes then why?

From the background I read about the properties of integer $n=6$ ,I find that its satisfying many interesting properties , In particular in number theory , $n=6$ is the first perfect number and all ...
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### On odd perfect numbers and a GCD - Part II

Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
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### Attempt to get a characterization of even perfect numbers from an equation involving the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(n)$ for integers $n\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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### 'imperfect' numbers

A perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid. Recently, I noticed that at least for the initial integers, it is ...
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### Divisibility of perfect numbers

I am new at the Number theory, I have a question that; n is an even perfect number without 28, for all of the other even perfect numbers, prove that n = 1 or -1 (mod7). Actually, I don't know where ...
### Implications of $q \neq 1049$ when $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special / Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, we let $$I(x) = \frac{\sigma(x)}{x}$$ denote the ...
### On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number
MOTIVATION Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2,$$ i.e., $q$ is the special / Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. ...