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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Follow-up to MSE question 3738458

This is a follow-up inquiry to this MSE question. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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Primes with prime digits and prime sum of the digits

I don't know if this concept is already defined, I consider the number $n$ to be a perfect prime (weak) if $n$ is a prime, its digits are primes and the sum of the digits is prime. Every one digit ...
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Can this inequality be solved for $q$ in terms of $n$ (or the other way around), if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. The following general inequality relating $k$ and $q$ is proved in this ...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
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On Carmichael function and aliquot parts of odd perfect numbers

We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael function dedicated to this number ...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs: $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \...
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3 votes
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Reference request regarding odd 4-perfect numbers

I am reading a paper by Broughan and Zhou (2006) that is dealing with odd 4-perfect numbers. The title of the paper is "Odd multiperfect numbers of abundancy four." In this paper they ...
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semi perfect number - number of divisors

The definition: we define number as semi perfect , if the number equals to the sum of exactly k of its divisors. the question: prove that for every n (n>0 | n belong to N) n is semi perfect order 3 ...
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On odd perfect numbers and a GCD - Part VIII

(This question is an offshoot of earlier posts with a similar title and this recent preprint.) Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv ...
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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

The following query is an offshoot of this post 1 and this post 2. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect numbers likely ...
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263 views

What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.) A number $P$ is said to be perfect if $\...
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Is there a connection between perfect numbers and perfect graphs? [closed]

In graph theory, a perfect graph "is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number)". In number ...
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I have found a proof that there does not exist any odd perfect number, Am I Correct? [closed]

To Verify the proof click here I really cannot imagine I did it, My Intuition to send this here is just please God there must be a mistake in here, but if magically there is isn't, I will really say ...
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Perfect numbers and Pell's equation

(Disclaimer: The following is a naive attempt to apply the theory of Pell's equations to perfect numbers. Please bear in mind that this is my first time to try solving such an equation.) Let $p^k$ be ...
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On odd perfect numbers and a GCD - Part VII

(Pardon me for being somewhat stubborn, but this question will be the last for this week. This post is an offshoot of this one.) Let $N = q^k n^2$ be an odd perfect number be an odd perfect number ...
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Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?

This question is an offshoot of this post #1 and this post #2. Let $N = q^k n^2$ be an odd perfect number be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\...
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Please check my proof: $\gcd(\sigma(q^k),\sigma(n^2)) \mid \gcd(n^2,\sigma(n^2))$ holds if $q^k n^2$ is an odd perfect number with special prime $q$.

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)\cdot\gcd\bigg(n^2,\...
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Please check my proof: If $q^k n^2$ is an odd perfect number with special prime $q$, then $\sigma(q^k)/2 \mid n$ if and only if $n \mid \sigma(n^2)$.

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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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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4 votes
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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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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A number $N$ is said to be perfect if $\sigma(N)=...
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3 answers
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Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
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1 answer
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

This post is an offshoot of this MSE question. Motivation Let $x, y$ and $z$ be positive integers. Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\...
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If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?

Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. A number $N$ is said to be perfect if $\sigma(N)=2N$. ...
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1 answer
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On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$ - Part II

Hereinafter, 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 $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ ...
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3 answers
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On the conjectured inequality $q > k$, where $q^k n^2$ is an odd perfect number with special prime $q$

Let $N=q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ ...
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1 answer
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On odd perfect numbers and a GCD - Part VI

(Note: This post is closely related to this earlier MSE question.) Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$....
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2 votes
1 answer
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On odd perfect numbers and a GCD - Part V

(Note: This post is tangentially related to this earlier MSE question.) Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,...
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3 votes
2 answers
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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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Is it true that $l_1(q,n) \geq g(k)$, if $q^k n^2$ is an odd perfect number with special prime $q$?

(Note: This post is an offshoot of these earlier questions: (post 1) and (post 2).) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(...
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1 vote
1 answer
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On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number

(Note: This post is an offshoot of this earlier MSE question.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the ...
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1 answer
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Elementary Proof of No Odd Perfect Numbers [closed]

I came across this proof on the Arxiv that there are no odd perfect numbers. It is elementary and easy to follow and looks correct to me? Of course there must be a mistake there somewhere but I am not ...
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Is it possible to improve on these bounds for $\frac{\varphi(n)}{n}$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the Euler-totient function of the positive integer $x$ by ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is $\gcd(\sigma(q^k),\sigma(n^2))=1$ equivalent to $k=1$?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Here is my: QUESTION If $q^k n^2$ is an odd perfect number with special prime ...
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2 votes
1 answer
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On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part III

(Preamble: This question is an offshoot of this answer by mathlove to an earlier post.) Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of the divisors of the positive integer $x$. If $\sigma(m)=...
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2 votes
2 answers
77 views

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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1 vote
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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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1 vote
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part VI

(Note: This question has been cross-posted to MO.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\...
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3 votes
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Show that if $n$ is a positive integer greater than $1$, then the Mersenne number $M_n$ cannot be the power of a positive integer.

enter image description here This is my progress. How do I go ahead? Transcription of image above Let $M_n = 2^n - 1$ be a Mersenne number. Suppose that $M_n = q^k$ for some $q \in \mathbb{N}$. $M_n$ ...
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1 answer
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Proof verification - Discarding a subcase of odd perfect numbers

Thinking about the problem of the existence of odd perfect numbers, I elaborated the following ideas. I post them here for you to check if they are correct, or point out the errors. We define the sum ...
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part V

(Preamble: This post is an offshoot of this MSE question and this MO question.) My primary aim in this post is to compute a (hopefully factorable) expression for the quantity $n^2 - q^k$, if $N = q^k ...
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0 votes
1 answer
125 views

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)...
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0 votes
2 answers
269 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part IV

(Preamble: This post is an offshoot of this earlier MSE question.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\...
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2 votes
1 answer
144 views

On odd perfect numbers and a GCD - Part IV

This question is related to this post1 and this post2. Hereinafter, we will let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$, and $\gcd(y,z)$ will denote ...
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3 votes
1 answer
104 views

How to find all even perfect number of a particular form. [closed]

Find all the even perfect numbers of the form $a^a +1$, where $a \in \mathbb{N}$. Can you please provide me some hint or idea on how to find such numbers.
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2 votes
1 answer
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How do number shapes relate to k-perfect numbers?

We know that all perfect numbers are a Mersenne prime, multiplied with the corresponding power of 2 for that prime, and then halved.$$2^{n}-1(2^{n-1})$$ It is also true that all perfect numbers are ...
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1 vote
1 answer
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On the inequality $I(n^2) \geq (6 - s(q^k))/3$, where $q^k n^2$ is an odd perfect number and $I(x)$ is the abundancy index of $x$

In what follows, we denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, the aliquot sum of $x$ by $s(x)=\sigma(x)-...
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2 votes
1 answer
130 views

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 ...
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