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.

273 questions
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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 ...
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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 ...
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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 ...
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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 $825$.
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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 $2^p-1$...
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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 ...
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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).
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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 ...
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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 ...
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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 ...
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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: ...
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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,...
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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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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 ...
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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 ...
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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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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: ...
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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 ...
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 ...