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.

17
votes
1answer
286 views

When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

Let $\sigma(x)$ denote the sum of the divisors of $x$, and denote the abundancy index of $x$ as $$I(x) = \dfrac{\sigma(x)}{x},$$ and the deficiency of $x$ as $$D(x) = 2x - \sigma(x).$$ If the equation ...
16
votes
2answers
2k views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
16
votes
2answers
687 views

If $N = q^k n^2$ is an odd perfect number and $n < q^{k+1}$, does it follow that $k > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $49$ known examples of even perfect numbers -- on ...
12
votes
1answer
881 views

Can an odd perfect number be divisible by $825$?

I know that an odd perfect number cannot be divisible by $105$. I wonder if that's also the case for $825$.
12
votes
1answer
426 views

How to show that all even perfect numbers are obtained via Mersenne primes?

A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). A well known theorem by Euler states that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$...
11
votes
3answers
3k views

Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number ...
11
votes
1answer
632 views

Can an odd perfect number be divisible by $165$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
10
votes
1answer
210 views

On odd perfect numbers $n$ and $\sigma\left(n^\lambda\right)$

As background, I don't know if this kind of calculations were in the literature and/or are interestings. I can to prove that being $\lambda\geq 1$ a fixed integer, $n$ is perfect if and only if $$2n=\...
10
votes
1answer
358 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...
9
votes
1answer
177 views

Is $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ a square number for integer $x>1$

Motivated by this question. How to prove that $\left( {{2}^{x}}-1 \right)\left( {{5}^{x}}-1 \right)$ is not a square number for integer $x>1$? Thanks for any suggestions. Edition by the ...
9
votes
1answer
307 views

How did Descartes come up with the spoof odd perfect number $198585576189$?

We call $n$ a spoof odd perfect number if $n$ is odd and and $n=km$ for two integers $k, m > 1$ such that $\sigma(k)(m + 1) = 2n$, where $\sigma$ is the sum-of-divisors function. In a letter to ...
9
votes
0answers
289 views

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-\...
8
votes
1answer
779 views

Can an odd perfect number be divisible by $105$?

I have a tough one today. Show that if $n$ is an odd perfect number, then not all of $3$, $5$, and $7$ are divisors of $n$. Any and all help is appreciated. Thanks very much.
7
votes
2answers
2k views

Discussion on even and odd perfect numbers.

First of all thank you so much for answering my previous post. These are few interesting problems drawn from Prof. Gandhi lecture notes. kindly discuss: 1) If $n$ is even perfect number then $(8n +1)$...
7
votes
1answer
131 views

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\...
6
votes
3answers
1k views

If $n$ is an even perfect number $ n> 6$ show that the sum of its digits is $\equiv 1 (\bmod 9)$

If $n$ is an even perfect number $ n> 6$ show that the sum of its digits is $\equiv 1 \mod 9$. I know perfect numbers are of the form $(2^{p-1})(2^{p}-1)$. I have a few trials that I have done and ...
6
votes
2answers
108 views

Median divisor of even perfect numbers

I noticed that when divisors of even perfect numbers are listed in ascending order, the middle divisor (I guess the median), is always of the form $2^n$, some power of 2. If true is there a proof for ...
6
votes
3answers
3k views

Can powers of primes be perfect numbers?

I need to prove the following, though I'm not 100% certain I understand the definition of a perfect number. Prove that no perfect number is a power of a prime. First of all, I'm assuming that the ...
6
votes
2answers
6k views

Applications of Perfect Numbers

I'm preparing a talk on Mersenne primes, Perfect numbers and Fermat primes. In trying to provide motivation for it all I'd like to provide an application of these things. I came up with these: ...
6
votes
2answers
161 views

Can a multi-perfect number be a perfect square?

It is fairly easy to show that a perfect number $\Gamma$ cannot be written in the form $\Gamma=n^2$ for integer values of $n$. However, does this property hold true for multi-perfect numbers---that is,...
6
votes
0answers
61 views

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 ...
6
votes
0answers
89 views

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, $n$ $=$ $...
5
votes
3answers
134 views

Prove that if $n$ is a perfect number, $kn$ is not

Prove that if $n$ is a perfect number, $kn$ is not. If $\gcd(k,n)=1$ then this is clear. (assume $\sigma(n)=2n$ , $\sigma(nk)=2kn$ , then $k=\sigma(k)$ but $\sigma(k)>k)$. But what about $\...
5
votes
2answers
886 views

Is this proof that there are no perfect, odd, integer square numbers legitimate?

Assumptions: Any even number times any other number is always an even number. An odd number times an odd number is always an odd number. An even number plus an even number is even, and an odd number ...
5
votes
1answer
2k views

Super Perfect numbers

A super-perfect number is a number with $\sigma(\sigma (n))=2n$. How can I prove that every even super perfect number is from the form $n=2^k$ when $2^{k+1}-1$ is prime. I tried every way please ...
5
votes
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 ...
5
votes
1answer
175 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since $37=(...
5
votes
1answer
95 views

Is the odd part of even almost perfect numbers (other than the powers of two) not almost perfect?

Let $\sigma(x)$ denote the sum of the divisors of $x$. A number $M$ is called almost perfect if $\sigma(M) = 2M - 1$. If $M$ is an even almost perfect number, then the only known examples for $M$ ...
5
votes
1answer
981 views

Proving that a number is perfect iff the sum of the reciprocal of its divisors is $1$

I am trying to prove the following theorem: Theorem. A number is perfect iff the sum of the reciprocal of its divisors, excluding $1$, is $1$. Thus far, this is the proof that I have managed to ...
5
votes
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 ...
5
votes
1answer
80 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, does $I(n^2) \geq 5/3$ imply $k=1$?

Let $\sigma(x)$ denote the sum of the divisors of $x \in \mathbb{N}$. Denote the abundancy index of $y \in \mathbb{N}$ by $I(y)=\sigma(y)/y$. If $\sigma(N)=2N$, then $N$ is said to be perfect. ...
5
votes
0answers
108 views

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 ...
4
votes
2answers
723 views

Prove that any power of a prime is not a perfect number [closed]

How do I prove: Let $p$ be a prime, and $n$ be a positive integer. Then $p^n$ is not a perfect number. One example is when $p = 2$ and $n = 3$, the question is to show $8$ is not a perfect number....
4
votes
3answers
3k views

A problem dealing with even perfect numbers.

Question: Show that all even perfect numbers end in 6 or 8. This is what I have. All even perfect numbers are of the form $n=2^{p-1}(2^p -1)$ where $p$ is prime and so is $(2^p -1)$. What I did was ...
4
votes
2answers
275 views

Origin of Almost Perfect Numbers

Let $N$ be a positive integer. $N$ is called a perfect number if the sum of its positive divisors denoted by $\sigma(N)=2N$. For example $6$ is a perfect number since: $\sigma(N)=1+2+3+6=12=2(6)$. ...
4
votes
1answer
111 views

Are perfect numbers definable in $(\Bbb N, 0, 1, +, \cdot)$?

Let $\mathscr L$ be the first-order language having two symbols of constants $\underline{0},\underline{1}$ and two symbols of binary operations $\underline{+},\underline{\cdot} \,.\,$ We consider the $...
4
votes
1answer
37 views

On an interesting assertion in the OeisWiki page on multiply-perfect numbers

The following (interesting) assertion appears in the OeisWiki page on multiply-perfect numbers: ...
4
votes
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}...
4
votes
1answer
536 views

Relationship between Mersenne Primes and Triangular / Perfect Numbers

I'm a new user and have only a college sophomore's understanding of mathematics, so please bear with me. I was reading a book titled “The Simpsons and their Mathematical Secrets” in which the author ...
4
votes
2answers
224 views

Can an odd perfect number be divisible by $5313$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $5313$.
4
votes
1answer
147 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 ...
4
votes
1answer
97 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers

(Note: This question has been cross-posted from MO.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M ...
4
votes
1answer
119 views

A property for perfect numbers

If $n=\prod\limits_{i=1}^k p_i^{\alpha_i}$, is a perfect number, with $p_i$ distinct primes, $\alpha_i\geq1$, then for each $i$ there is a $j \in \{1,\ldots,k\}$ such that $p_i \mid (p_j^{\alpha_j+1}-...
4
votes
0answers
88 views

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 ...
4
votes
1answer
285 views

Is it possible to simplify this expression even further?

(Preamble: This question is tangentially related to this earlier one.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $...
4
votes
0answers
108 views

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 ...
4
votes
0answers
148 views

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 ...
3
votes
1answer
164 views

On the undecidability of the existence of odd perfect numbers

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. That is, $$\sigma(x) = \sum_{d \hspace{0.01in} \mid \hspace{0.01in} x}{d}.$$ For example, the divisors of $28$ are $$1, 2, ...
3
votes
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 ...
3
votes
2answers
78 views

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 ...