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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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On $\text{Lower bound}\leq \operatorname{rad}(n)$, where $n$ is an odd perfect number: reference request or what work can be done about it

For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ ...
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2answers
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Solve for $p$ in this equation $\;2^{p−1}(2^p − 1) = X$.

Solve for $p$ in this equation $2^{p−1}(2^p − 1) = X$. This is a general formula for finding $X$ (an even perfect number), where $p$ is any prime number. I want to find $p$ when $X$ is given, so that ...
2
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1answer
124 views

Could a Mersenne prime divide an odd perfect number?

The relationship between Mersenne primes $2^r-1$ and even perfect numbers $2^{r-1}(2^r-1)$ is well-known (Euclid, Euler). In a video on the web I heard the statement that it is known that a Mersenne ...
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1answer
56 views

On a conjecture that $P_n^{\,2}+5^2+2^k=(P_n-1)^2+l^2$.

I was looking at perfect numbers and came across something that might serve a little interesting. Denote by $P_n$ the $n^\text{th}$ perfect number, then there appears to always exist $k\in\mathbb{W}...
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2answers
49 views

Would like to get numerical (lower [and upper?]) bounds for $p$

This question is an offshoot of this earlier MSE question. Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the abundancy index of $z$ by $I(z) :...
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1answer
64 views

Confirmation of Proof: $\pi_n + p_a + p_b \geqslant \sum_{i=1}^4 x_i$.

I was messing around with numbers and I made the following conjecture: Conjecture: Let $\pi_n$ be the $n^{\text{th}}$ perfect number; $p_a$ be the prime after $\pi_n$ and $p_b$ be the ...
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100 views

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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1answer
113 views

Odd perfect numbers, questions on a proof of Steuerwald

I'm currently working on odd perfect numbers, to be precise on a proof (german language) of Rudolf Steuerwald. I have two questions regarding this. Euler proved that any odd perfect number $n$ has to ...
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82 views

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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1answer
55 views

A question on entropy of natural numbers

This question is more of a conjecture, but I am not posting it to mathoverflow, since I am not a professional mathematician and I do not know if it is research related: Let $n=p^{a} s$, $p$ be the ...
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1answer
85 views

Equations involving arithmetic functions, totatives and even perfect numbers

I've deduced simple relationships that satisfy each even perfecf number (even numbers $n$ for which $\sum_{d\mid n}d=2n$) and now I wondered about related conjectures. For each integer $m\geq 1$ we ...
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35 views

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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1answer
72 views

If $\sigma(n)=2n$, how about the relationship between $\sigma(mn)$ and $2mn$?

The question simply states "Prove that any multiple of a perfect number is abundant." Perfect number and Abundant number are each defined by $\sigma(n)=2n$ and $\sigma(n)>2n$ How to solve this ...
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1answer
29 views

Does there exist a natural number $N = xy$ satisfying $D(x)D(y) = 2s(x)s(y)$ with $\gcd(x,y)=1$?

Preamble: I apologize in advance if what I am asking for in this question, I could get an answer easily so myself, for example by coding a short Mathematica script. It is just that I have not yet ...
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103 views

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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1answer
78 views

Any proof for :Sum of two perfect number never be a perfect number?

I want to know if there is a proof to show that Sum of two perfect number never be a perfect number by the way if there are a finitely many perfects numbers which they are sum of two perfect numbers ?...
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1answer
61 views

On miscellaneous questions about perfect numbers III

This is a wild guess about odd perfect numbers. Thus you can see it as an exercise and not as a serious conjecture. I add here the MathWorld's reference dedicated to odd perfect numbers. Question. ...
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If $q^k n^2$ is an odd perfect number with Euler prime $q$, does this equation imply that $k=1$?

Let $\sigma(x)$ be the sum of the divisors of $x$. Denote the deficiency of $x$ by $D(x) := 2x - \sigma(x)$, and the sum of the aliquot divisors of $x$ by $s(x) := \sigma(x) - x$. Here is my ...
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1answer
99 views

Is the occurence of two perfect numbers a coincidence?

The expression $$n^n+n!+1$$ is prime for the following non-negative integers $n\le 7\ 600$ : $$[0, 1, 2, 4, 28, 496]$$ if we assume $0^0=1$ The numbers $28$ and $496$ are perfect numbers. Is ...
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1answer
95 views

Is there a $31$-dimensional manifold with 496 differential structures?

Milnor found a $7$-dimensional sphere with 28 differential structures. Is there a $31$-dimensional manifold with 496 differential structures?
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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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93 views

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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133 views

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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1answer
65 views

If an odd perfect number exist could be a solitary number? [closed]

Perfect numbers is a number that is half the sum of all of its positive divisors .And solitary numbers means that $\frac {\sigma(n)}{n}$ is an irreducible fraction, it's seems to me that all even ...
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25 views

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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1answer
33 views

Identify sequences from OEIS or the literature, or find examples of odd integers $n\geq 1$ satisfying these equations related to odd perfect numbers

Let $\sigma(m)=\sum_{d\mid m}d$ the sum of divisors function, and $\varphi(m)$ the Euler's totient function, then it is possible to prove the following statements. And I would like to identify some ...
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1answer
44 views

A conjecture for an inequality for every odd number.

Let $E(n) = \sum_{d|n} d \log(d) $. Then for coprime $n,m$ we have $$E(mn) = \sigma(m) E(n) + \sigma(n) E(m).$$ Suppose there exists an odd perfect number $n$. Then$$E(2n) = E(2)\sigma(n)+E(n)\sigma(2)...
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1answer
114 views

Is it true that every odd perfect number can be written in the form $\frac{r\sigma(r)}{2r - \sigma(r)}$?

(Note: This question has been cross-posted to MO.) Is it true that every odd perfect number $N = q^k n^2$ can be written in the form $$N = \frac{r\sigma(r)}{2r - \sigma(r)}?$$ (Here, $q$ is a prime ...
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239 views

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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1answer
37 views

A simple representation that satisfies every even perfect number: products over the squarefree parts of its divisors

I wrote a draft of next statement for even perfect numbers that I believe that isn't in the literatute. I am asking to know a rigorous and simple proof. Question. Prove that for each fixed even ...
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349 views

What Are The Practical Applications Of Perfect Numbers?

What are the practical applications of perfect numbers? When googling, all I found was practical applications of mersenne primes. If you can't find any (like I), are there any applications of abundant ...
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Confirmation of Proof: If $p$ is a perfect number, then $1 + 8p$ is a squared number.

I have created a hypothesis: $$\text{If } p \text{ is a perfect number, then } 1 + 8p \text{ is a squared number.}$$ Could anyone prove/disprove the above statement? I have no idea where to begin, ...
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2answers
294 views

Can you prove or disprove the following list of my conjectures?

The following three statements are my own conjectures, not a homework problem. $a)$ For $n = 3, 4, 5,..$, every square integer $n^2$ can be expressed as the sum of a prime $p$ and two other primes $...
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174 views

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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1answer
121 views

Descartes number

In 1638 Descartes wrote a letter to Mersenne where he talks about how the number $$D=3^2⋅7^2⋅11^2⋅13^2⋅22021$$ would be an odd perfect number if we mistakingly assume that $22021$ is prime. My ...
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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 ...
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2answers
323 views

“I am sure there are infinitely many perfect numbers”

The question Are there infinitely many perfect numbers? is a classic old unsolved problem. However, we keep finding perfect numbers (via Mersenne primes) and produce a lot of knowledge on perfect ...
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1answer
53 views

Does $P = \big\{(2^n - 1)^2 - \sum_{k = 1}^{2^{n - 1} - 1}(4k - 1) : P = 2^{n - 1}(2^n - 1) = \text{Perfect Number}\big\}?$

So recently I had figured out on my own that: $$1 + 2 + \cdots + n = P \iff 2^{n - 1}(2^n - 1) = \{P : P = \text{Perfect Number}\}$$ Now I had figured out something else as well: $$1 - 2^2 + \cdots - (...
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1answer
45 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, which of the following relationships between $q^2$ and $n$ hold?

(Preamble #1: In what follows, we take $\sigma=\sigma_{1}$ to be the sum of the divisors, and denote the abundancy index of $x \in \mathbb{N}$ as $I(x)=\sigma(x)/x$.) (Preamble #2: My sincerest ...
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1answer
211 views

Odd Perfect number does not exist

Odd perfect numbers do not exist. According to Euclid's formulae a perfect number is equal to $2^P(2^P-1$) where $2^P-1$ should be prime. Only $2$ is the even prime number and rest are odd. So $2^P-1$ ...
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1answer
33 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, does $\sigma(n^2)/q^k < q$ hold?

If $q^k n^2$ is an odd perfect number with Euler prime $q$, then it is known that $$\frac{\sigma(n^2)}{q^k}=\frac{2n^2}{\sigma(q^k)}=\frac{D(n^2)}{\sigma(q^{k-1})}=\gcd\left(n^2,\sigma(n^2)\right),$$ ...
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Show $n$ is perfect $\iff \sum\limits_{k=1}^{n-2}k\left\lfloor\frac{n}k\right\rfloor=1+\sum\limits_{k=1}^{n-1}k\left\lfloor\frac{n-1}k\right\rfloor$

So the question is: Prove that $n$ must be a perfect number $\iff$ $$\sum_{k = 1}^{n - 2}k\left \lfloor\frac{n}{k}\right \rfloor = 1 + \sum_{k = 1}^{n - 1}k \left \lfloor \frac{n - 1}{k}\right\...
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1answer
75 views

On the Descartes-Frenicle-Sorli conjecture and the Euler prime of odd perfect numbers

(Preamble: My apologies for the somewhat long post - I merely wanted to include all the details that I had in mind for ease of reference later.) This post is an offshoot of this earlier MSE question. ...
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1answer
65 views

Prove that $φ^x$ is close to an integer $y : x = 2^{n - 1}(2^n - 1)$

I have a question. I have recently discovered that if you get the golden ratio $φ$, which is equal to $\frac{1 + \sqrt 5}{2}$, then if you raise this to the power of $2^{n - 1}(2^n - 1)$, the higher ...
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1answer
146 views

Special prime of odd perfect number

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e.,$q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$ . (That is, $\sigma(N)=2N$ where $\sigma$ is the classical ...
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1answer
81 views

Do all terms $n$ of OEIS sequence A228059 have a $p$ with exponent $1$?

Do all terms $n$ of OEIS sequence A228059 have a $p$ with exponent $1$? OEIS sequence A228059: Odd numbers of the form $p^{1+4k}{r^2}$, where $p$ is prime of the form $1+4m$, $r > 1$, and $\gcd(p,...
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1answer
67 views

If $N=q^k n^2$ is an odd perfect number and $q = k$, why does this bound not imply $q > 5$?

Let $\mathbb{N}$ denote the set of natural numbers (i.e., positive integers). A number $N \in \mathbb{N}$ is said to be perfect if $\sigma(N)=2N$, where $\sigma=\sigma_{1}$ is the classical sum of ...
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1answer
78 views

A question on the Euler prime of odd perfect numbers

A number $N \in \mathbb{N}$ is said to be perfect if $\sigma(N)=2N$, where $\sigma=\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma(6)=1+2+3+6=2\cdot{6}$, so that $6$ is ...
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4answers
110 views

Show that even perfect numbers (other than 6) have a remainder 1 on division by 9

Every even perfect number can be written in the form, $2^{p-1}(2^p-1)$ so this means I want to show that $2^{p-1}(2^p-1)\equiv1$ mod $9$ but can't see where to go from here.