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 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-\...
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Does there exist an integer $m$ which is not in the image of $D(n)=2n - \sigma(n)$?

Let $\sigma(n)$ be the sum of the divisors of the positive integer $n$, and denote the deficiency of $n$ by $D(n)=2n-\sigma(n)$. Here is my question: Do there exist numbers $m$ which (provably) do ...
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Any value $n$ we can conclude there are no perfect numbers of the form $kn+1$ or congruent to $1$ $\pmod n$?

Is it proven for any integer $n$ that there are no perfect numbers of the form $kn+1$ or more simply it has been proven there are no perfect numbers congruent to $1$ $\pmod n$? For instance, $n$ $=$ $...
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Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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Is there a notion “above” that of perfect numbers?

When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea. A basic ...
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If we suppose, that a perfect odd number exists, can we determine whether infinite many exist?

It is currently not known whether odd perfect numbers (numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$, including $1$ and $n$) exist. But suppose, a perfect ...
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If $q^k n^2$ is an odd perfect number with Euler prime $q$, can $q=17$ hold?

Note: This question is an offshoot of this earlier MSE post. If $N$ is odd and $\sigma(N)=2N$ where $\sigma=\sigma_{1}$ is the classical sum-of-divisors function, then $N$ is said to be an odd ...
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If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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On some inequalities relating the special/Euler prime and non-Euler part of odd perfect numbers

Let $N$ be an odd (positive) integer. If $\sigma(N)=2N$ where $\sigma(N)$ is the sum of the divisors of $N$, then $N$ is called an odd perfect number. Let $I(N)=\sigma(N)/N$ denote the abundancy ...
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On Basak's “Bounds On Factors Of Odd Perfect Numbers”

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, we denote the ...
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Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
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<Reference Request> Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This has been cross-posted to MO.) Good day! I would like to request for references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor. ...
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Are there any Perfectly Abundant Numbers?

Define a Perfectly Abundant Number as a number with the property: $$ x : \sigma(x) = 2x + 1, $$ where $\sigma$ is the sum of the factors of $x$, including itself. In other words, $x$ isn't a perfect ...
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Can an odd perfect number be divisible by $101$?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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On prime factors of odd perfect numbers

Why is it so difficult to determine actual numerical values for prime factors of odd perfect numbers? Recall that, if $N$ is an odd perfect number, then Euler proved that it takes the form $N = q^k n^...
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Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...
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Are there any other squares $n^2$ for which $\gcd(n^2, \sigma(n^2)) = 2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. Denote the deficiency of $x$ by $$D(x)=2x-\sigma(x).$$ I am interested in solutions to the equation $$\gcd(n^2, \sigma(n^2)...
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Inquiry about the Wolfram MathWorld entry on Odd Perfect Number

I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number. Last sentence of the fifth paragraph states that: Hagis (1980) showed that odd perfect numbers must have at ...
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Equations involving the Euler's totient function and Mersenne primes

In this post we denote the Euler's totient function as $\varphi(n)$, first we show a claim related to Mersenne primes, see for example this Wikipedia and secondly we are going to ask a related ...
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Can the following argument be pushed to a full proof that $(p + 2)/p$ is an outlaw if $p$ is an odd prime?

This is related to this earlier MSE question. Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $I(x):=\sigma(x)/x$. If the equation $I(a) = b/c$ has no ...
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Even perfect number that is also a sum of two cubes

In 2010, Gallardo proved that the only even perfect number that is also a sum of two cubes is $28$. Here is a link to his proof (see the second page). The first part goes roughly as follows: Let $N$ ...
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Does the existence of infinite number of Leinster groups indicate the existence of infinite number of perfect numbers?

A Leinster group is a finite group $G$, such, that that the sum of orders of its normal subgroups is $2|G|$. A perfect number is a positive integer that is equal to the sum of its proper positive ...
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What about the equation $\sigma(2n)=2\left(n+\sigma(n)\right),$ involving the sum of divisor function?

Yesterday I wrote this equation involving the sum of divisors function $\sigma(l)=\sum_{d\mid l}d$, $$\sigma(2n)=2\left(n+\sigma(n)\right).\tag{1}$$ Due to its very simple form I don't know if it is ...
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Are there imperfects or pluperfects numbers?

I am not a mathematician nor a cientist, I'm just a curious person. My math background is always trying to be "back there", as anything you learn tends to be. So, there is a risk that I post silly ...
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Assuming the Lenstra–Pomerance–Wagstaff conjecture, in what range of values for $p$ would we expect to find the next prime $2^p - 1$?

Assuming the Lenstra–Pomerance–Wagstaff conjecture is correct, in what range of values for a prime $p$ would we expect to find the next Mersenne prime $2^p - 1$?
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On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

(Note: This question has been cross-posted to MO.) Let $\sigma$ be the classical sum-of-divisors function. For example, $$\sigma(6)=1+2+3+6=12={2}\cdot{6}.$$ If $\sigma(N)=2N$ and $N$ is odd, then $...
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On identities and congruences involving the harmonic mean of odd perfect numbers

I would like to know if my calculations were rights and if you want to deduce other different interesting congruences following my approach. On assumption (notice that it isn't the usually accepted ...
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Can an odd perfect number be divisible by either $2049$ or $2051$?

Can an odd perfect number be divisible by either $2049$ or $2051$? Note that $2049 = 3 \cdot {683}$, and that $2051 = 7 \cdot {293}$. Added July 15 2016 It is known that an odd perfect number ...
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A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) $$1^2+2^2+\ldots+n^2=\frac{n(n+1)(...
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Requesting for details of Grun's result on odd perfect numbers

This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun $1952$ proved that the smallest prime factor $p_1$ (of an odd perfect number $N$) is $< \frac{2\...
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Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...
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If n>6 is an even perfect number, Prove that n is congruent to 4 (mod12)

I know that for n to be an even perfect number greater than 6 it has the form $ 2^(m-1)(2^m-1)$ where m is prime. I also know that since n is an even perfect number, it is congruent to 1 (mod 9). ...
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With $rad(N)=\prod_{p|N}p$, if $N$ is even and $\frac{2+rad(N)}{8}\left(\sum_{\substack{d|N,d<rad(N)}}d\right)=N$ then is perfect?

In the literature (see for example sites and paper concerning to the abc conjecture, I say this as reference and by caution to avoid mistakes) is defined the arithmetical function $rad(n)$ as $1$ if $...
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Can an odd perfect number be divisible by $119$?

We know that an odd perfect number cannot be divisible by $105$ (see here), by $825$ (see here), and by $5313$ (see here). I wonder if that's also the case for $119$.
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On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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What do we know about Sorli's Conjecture and Odd Perfect Numbers?

I have seen that Jose Arnaldo Dris has recently published a great deal of literature concerning odd perfect numbers and Sorli's Conjecture. He claims that the truth of Sorli's conjecture implies that ...
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Combinatorial interpretation of Euclid's form for even perfect numbers

Euclid showed that if $p$ is a prime such that $2^{p}-1$ is also a prime, then the number $n=2^{p-1}.(2^{p}-1)$ is perfect. Much later, Euler proved that every even perfect number is of this form. ...
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Did Descartes and Frénicle consider deficient-perfect numbers?

Holdener and Rachfal show that, if $N = q^k n^2$ is an odd perfect number with special prime $q$ (satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), then $q^{(k-1)/2} n^2$ is deficient-...
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Does this iterated sequence always end in a finite number of steps to a number which is divisible by a perfect number?

I posted this question at MathOverflow, but then I realized that maybe it is more appropriate to ask it here: Let $f$ be a multiplicative arithmetic function which maps $\mathbb{N}$ to itself, such ...
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On GCDs and odd perfect numbers

Let $N=q^k n^2$ be an odd perfect number with special prime $q$. The index $i(q)$ of $N$ at the prime $q$ is then equal to $$i(q):=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/2}=\frac{D(n^2)}{s(q^...
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A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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Divisors of a $ k $-multiperfect number

Let $ n $ be a $ k $-multiperfect number. Denote by $ d_m $ its $ m $ smallest divisor, and $ n_{m} $ the number of divisors of $ n $ divisible by $ d_m $. Is there for all $ 2\leq m\leq\tau(n) $...
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Has it been conjectured that all $k$-multiperfect numbers are multiples of $k$?

A quick glance at the list of the first $ k $ -multiperfect numbers for small $ k $ makes me think that all $ k $ -multiperfect numbers are multiples of $ k $ , which is a generalization of the ...
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Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd ...
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If $n = km$ is a Descartes number with quasi-Euler prime $m$, then $m < k$.

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 numbers or positive integers. Recall that a Descartes number is an ...
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Bounds for an expression involving the divisors of an odd perfect number

Let $$\sigma(x) = \sum_{l \mid x}{l}.$$ That is, let $\sigma(x)$ denote the sum of the divisors of $x \in \mathbb{N}$ (the set of natural numbers or positive integers). Set $$D(x) := 2x - \sigma(x)$$ ...
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What is the relation between the perfect squares and prime numbers?

$\hspace{10pt}$I was playing with R and a list of the first 10,000 primes and suddenly I had the idea of looking how many perfect squares are between $p_n$ and $p_{n+1}$. $\hspace{10pt}$I make a ...
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Conjecture: $a^3 + b^3 + c^3 = p^3 \Rightarrow x^3 + y^3 + z^3 = \big(\frac{a + b + c}{2}\big)^3$

Conjecture: Let $p$ be an even perfect number, and $a, b, c$ be positive natural numbers. There exists values for $a, b, c$ to satisfy the following equation $$a^3 + b^3 + c^3 = p^3$$ for ...
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Does the product of two neighbouring primes hold a special property apart from always being semi-prime?

So I have made a theorem: You are given two prime numbers $P_n$ and $P_{n + 1}$ such that $P_n$ denotes the $n^{th}$ prime number and $P_{n + 1}$ is the following prime number from $P_n$. Take ...