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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If $q^k n^2$ is an odd perfect number with special prime $q$, does $q^k < n$ imply that $\sigma(q^k) < n$?

The topic of odd perfect numbers likely needs no introduction. Here is the: PROBLEM If $q^k n^2$ is an odd perfect number with special prime $q$, does $q^k < n$ imply that $\sigma(q^k) < n$? $\...
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Which primes have product equal to their sum plus 1?

Mathematically speaking, the question is to find all primes $p_i$ such that $$p_1p_2p_3\cdots p_k = 1+p_1+p_2+\cdots p_k$$ for some positive integer $k$. I know that the only solutions are $2$ and $3$,...
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The 'Factor chain' concept

I have been working on Odd Perfect Numbers for a while now. When I started to go through recent publications, I saw that a large number of papers have been proving results using the Factor Chain ...
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Two inequalities for proving that there are no odd perfect numbers?

Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}| d|n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}|d^2 | n\}$ be ...
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Is this function multiplicative and if so what is its value at prime powers?

For odd numbers $n$ let: $$a(n) = \sum_{d^2|n} d \frac{\sigma^*(n/d^2)}{2^{\omega(n/d^2)}}$$ where $\sigma^*(k) = $ sum of unitary ($\gcd(d,k/d)=1$) divisors of $k$ and $\omega$ counts the prime ...
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On the golden ratio and even perfect numbers

(Note: This post is an offshoot of this earlier MSE question.) Here is my question in this post: Is $I(2^{p-1}) - 1 > 1/I(2^{p-1})$ true when $I(2^{p-1}) = \sigma(2^{p-1})/2^{p-1}$ is the ...
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Can a multiperfect number be a perfect power?

(Note: The following post is an offshoot of this earlier MSE question: Can a multiperfect number be a perfect square?.) Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$...
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Is this a valid proof for $I(n^2) \geq \frac{5}{3}$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$, and let $I(x)=\sigma(x)/x$ be the abundancy index of $x$. Note that both $\sigma$ and $I$ are multiplicative functions. A number ...
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On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$

(Note: This post is an offshoot of the following earlier question.) Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the ...
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Are there numbers that have loop of 3 or more when adding their factors?

Are there numbers that have loop of 3 or more when adding their factors? I know perfect numbers when you add all their factors you get themself (I like to think of this everything important to them ...
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On the abundancy index of divisors of odd perfect numbers and a possible upper bound for the special/Euler prime

(Note: This post is an offshoot of this earlier question.) The topic of odd perfect numbers likely needs no introduction. Denote the sum of divisors of the positive integer $x$ by $\sigma(x)$, and ...
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Does $k=1$ follow from $I(5^k)+I(m^2) \leq \frac{43}{15}$, if $p^k m^2$ is an odd perfect number with special prime $p=5$?

The topic of odd perfect numbers likely needs no introduction. Denote the sum of divisors of the positive integer $x$ by $\sigma(x)$, and denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. Euler ...
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Reference request - Online copy of T. N. Sinha's “Note on perfect numbers”?

The topic of perfect numbers likely needs no introduction. Reference Request I was wondering if there is an online copy of the following journal article: T. N. Sinha, Note on perfect numbers, Math. ...
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On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$ - Part II

(Note: This question is a sequel to this earlier post.) Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where $$\sigma(x)=\sum_{d \mid x}{d}$$ is the sum of divisors of the ...
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If $N = q^k n^2$ is an odd perfect number with special prime $q$, then can $N$ be of the form $q^k \cdot (\sigma(q^k)/2) \cdot {n}$?

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 $28$ are perfect since $$\sigma(6) = 1 + 2 + 3 + 6 = ...
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On odd perfect numbers and a GCD - Part III

(Note: This post is an offshoot of this earlier MSE question.) In what follows, we let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. We also let $D(x)=2x-\sigma(x)$ denote the ...
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The relation with Perfect numbers , Triangular Numbers and Cyclops numbers

Triangular Numbers are numbers going from $1,3,6,10,15,21,28$ and so on . These numbers can be formed like a triangle , so called triangular numbers. Perfect numbers are numbers going in the pattern $...
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Suppose $n$ is an odd perfect number then it exists $p$ such that $\frac{n}{p}$ is a square

Suppose $n$ is an odd perfect number. How to show that it exists a prime number $p$ such that $\frac{n}{p}$ is a square number? My idea was: $n$ is perfect if $\sigma(n)=2n$. Let $n=2k+1, \ k \...
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An unusual pattern associated with perfect number

The pattern is that if you take a perfect number and place 5 at its unit digit. Then you have to square that number after squaring it the digits of the obtained number can be rearranged into two or ...
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Question about a result on odd perfect numbers - Part II

(This question is an offshoot of this earlier one.) In the paper titled Improving the Chen and Chen result for odd perfect numbers (Lemma 8, page 7), Broughan et al. show that if $$\frac{\sigma(n^2)}{...
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Is every perfect number triangular?

Euler proved that every even perfect number is of the form $p(p+1)/2$ for $p$ a Mersenne prime, in particular it is equal to the $p$-th triangular number (the sum $1 + \ldots + p$) for some Mersenne ...
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Assume an odd perfect number exists could be it written as $x^3+y^3+z^3$?

Touchard $(1953)$ proved that an odd perfect number, if it exists, must be of the form $12k+1$ or $ 36k+9$(Holdener $2002 $) , A necessary condition for any number to be written as a cubic sum $x^3+...
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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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Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$?

Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$? I did modification to the Mersenne numbers (Even perfect numbers) foruma I put that formual to be the power of 2 have got : $2^{...
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Show that if n is an even perfect number then n is not the sum of two squares. [closed]

A perfect number is a positive integer that is equal to the sum of its proper divisors, and all perfect numbers are even. A sum of two squares is an integer that is the sum of two squares integers.
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Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition?

(Note: This question is tangentially related to this earlier one.) Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural ...
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On solutions of $\varphi(n)=\frac{1}{2n}\sum_{1\leq d\mid n}\varphi(dn)$, where $\varphi(m)$ denotes the Euler's totient function

I wondered if one can to get easily an answer for the following question (I have thought about the other direction $\Leftarrow$). I don't know if it is in the literature, please refer it in comments ...
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On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number

I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Now this post is cross-posted on ...
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Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?

The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions: An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated ...
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On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$

Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where $$\sigma(x)=\sum_{d \mid x}{d}$$ is the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by ...
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Summing Odd Fractions to One, and Odd Perfect Numbers

The title says it all. Question What exactly is the relationship between Egyptian/unit fractions with odd denominators, and odd perfect numbers? Motivation In a comment underneath the question ...
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If $p^k m^2$ is an odd perfect number with special prime $p$, then what is wrong about the following factor chain approach to proving $p \neq 5$?

Suppose that $n = p^k m^2$ is an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. That $n$ is perfect essentially means that $$\sigma(p^k)\sigma(m^...
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Problems 11.3, No. 14 (a) and (b) from page 237 of David M. Burton's “Elementary Number Theory” (7th Edition)

Problem Statement Prove that (a) Any odd perfect number $n$ can be represented in the form $n = pa^2$, where $p$ is a prime. (b) If $n = pa^2$ is an odd perfect number, then $n \equiv p \pmod 8$. My ...
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Enrique Santos L's “Proof that no odd perfect number exists”

Background Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$. Let $n$ be an odd perfect number given in the so-called Eulerian ...
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A question on logic (related to odd perfect numbers)

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. If $\sigma(x)=2x$, then $x$ is called a perfect number. An odd perfect number $n$ is said to be given in the so-called ...
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Possible relationship between non-divisors of odd perfect numbers and coefficients of corresponding cyclotomic polynomials?

A positive integer $n$ is called perfect if $\sigma(n)=2n$, where $$\sigma(n)=\sum_{d \mid n}{d}$$ is the sum of divisors of $n$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect ...
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On the inequality $m < p^k$ where $p^k m^2$ is an odd perfect number

This question is an offshoot of this earlier one and this other question as well. Let $n = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(...
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If $q^k n^2$ is an odd perfect number with special prime $q$, then its index at the prime $q$ is not a square.

Let $N=q^k n^2$ be an odd perfect number with special prime $q$. (That is, $q$ satisfies $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.) The index $i(q)$ of $N$ at the prime $q$ is then equal to $$...
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Decision problem: Existence of a perfect number m larger than a natural number n

I am currently having a look at the slides from my theoretical computer science lecture and I am having trouble to understand a claim made. According to the slides the language $L = \{ n \in \mathbb{...
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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 ...
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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 ...
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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 $\gcd(q,n)=1$. ...

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