# 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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### (Applied to OPN) Let $c$ and $d$ be integers, not both equal to zero. If $q$ and $r$ are integers such that $c = dq + r$, then $\gcd(c,d)=\gcd(d,r)$.

I was reading the following web resource, and found the following Lemma 8.1, which I think would be immensely useful for a computational task on odd perfect numbers (hereinafter abbreviated as OPN): ...
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### Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

This question is an offshoot of this MSE answer. Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.) If $\sigma(M) = 2M$...
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### Show that, there do not exist two consecutive perfect numbers.

Show that, there do not exist two consecutive perfect numbers I know that it is unknown whether there exist any odd perfect number(s) or not. Also, I know that all the even perfect numbers are ...
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### Help with "A Simpler Dense Proof regarding the Abundancy Index."

I'm reading Richard Ryan's article "A Simpler Dense Proof regarding the Abundancy Index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
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### If $k=1$, then the index of an odd perfect number $p^k n^2$ (as defined by Chen and Chen (2014)) is not squarefree.

In what follows, we let $\sigma=\sigma_1$ denote the classical sum-of-divisors function. Denote the deficiency function of the positive integer $x$ by $D(x)=2(x)-\sigma(x)$, and the aliquot sum of $x$...
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### On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$

(Preamble: This question is an offshoot of this earlier MSE post.) Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. ...
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### Will it be possible to compute a factored expression for $n^2 - q^k$, if $q^k n^2$ is an odd perfect number with special prime $q$?

In what follows, we denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x)=\sum_{d \mid x}{d},$$ and the abundancy index of $x$ by $I(x)=\sigma(x)/x$. If $N$ is ...
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### On a possible proof for $p^k < m$, if $p^k m^2$ is an odd perfect number with special prime $p$

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. If $\sigma(M)=2M$ and $M$ is odd, then $M$ is called an odd perfect number (hereinafter abbreviated as OPN)...
### If an odd perfect number exists, does it have exactly one prime factor of the form $4a+1$?
I know that if an odd perfect number exists it must be of the form $p^kQ^2$ with $\gcd(p, Q) =1$ and $p \equiv k \equiv 1 \mod 4$. Reading the book The man who only loved numbers by Paul Hoffman, it ...