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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.

A positive integer $n$ is said to be a perfect number if it is equal to the sum of its proper positive divisors.

The smallest example of a perfect number is $6$ as it has positive proper divisors $1$, $2$, $3$, and $1 + 2 + 3 = 6$.

More generally, $2^{p-1}(2^p-1)$ is perfect whenever $2^p - 1$ is a prime (called a Mersenne prime); the case above corresponds to $p = 2$. Furthermore, every even perfect number is of this form.

It is currently unknown whether there are infinitely many perfect numbers or whether any odd perfect numbers exist.

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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 this, or does it happen all the time? I only…
Soulis
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Why are perfect numbers called perfect numbers?

A perfect number is a number than can be expressed as a sum of its factors. For example, 28 = 1 + 2 + 4 + 7 + 14 Why is this property important? What is so perfect about perfect numbers?
WorldGov
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Perfect numbers adding up to 10

In aliquot sequences, am I right in saying that all perfect numbers when you add up the digits the result always add up to $10$ ( apart from the perfect no $6$?) if so, is this just a coincidence? Example: "$496\mapsto 4+9+6=19\mapsto 1+9=10$".
Donna Skelly
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can't find the proof for the form of odd perfect numbers

I am currently researching perfect numbers and on the wikipedia page, as well as this paper: http://www.math.dartmouth.edu/~jvoight/articles/opn-mass-rev-060211.pdf it states that any odd perfect number N must be of the form: $$ N=q^{\alpha}…
maxG795
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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 questions here. I apologize to those that get a…
Nenzatti Bellece Nenzo
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Why is this inequality in Brent and Cohens paper on odd perfect numbers true?

In Brent and Cohen's paper about odd perfect numbers, they show this inequality. $N \ge p^a\sigma(p^a) \gt p ^ {2a}$ where a is even. I understand the next second half of this: $p^a\sigma(p^a) \gt p ^ {2a}$. The component * sum of its divisors is…
louis
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Unable to find next number of perfect numbers.

Perfect numbers are $6, 28, 496, 812, \dots $ From this source Here “double proportion” means that each number is twice the preceding number, as in $1, 2, 4, 8, ….$ For example, $1 + 2 + 4 = 7$ is prime; therefore, $7 × 4 = 28$ (“the sum multiplied…
Fawad
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Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to ${2^n}{-1}$.

Please explain in quotations! "Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to $2^n-1$." In a very simple case, for $n = 3, 1 + 2 + 4 = 7 = 8 - 1$.
William Zlacki
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For which natural numbers $n ≥ 999$ is the number $N = \sqrt{n-999}+\sqrt{n+1000}$ natural?

my question For which natural numbers $n ≥ 999$ is the number $N = \sqrt{n-999}+\sqrt{n+1000}$ natural? my idea I tried all the variants but it always gives that $n=999*1000$, which wont get $a$ and $b$ to be perfecr numbers
IONELA BUCIU
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Is any number one less than the sum of its proper divisors?

I've been fascinated by perfect numbers ever since I learned about them, but I don't think 1 should be counted in the sum. Every integer is divisible by itself and 1, so why is the number itself excluded and not 1? Are there any numbers that are the…
Cain Goldhardt
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Upper bounds of the frequency of perfect numbers.

What upper bounds exist for the frequency of perfect numbers. I.e. what functions $f(x)$ are there where we can say that as x tends to infinity, there exists so value n such that if $x>n$, the number of perfect numbers less than x is always less…
blademan9999
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Odd Perfect number does not exist: where is the error in this proof

I am a math aficionado unable to find out where is the error in this proof showing that a odd perfect number does not exist. May I get some help? Suppose $X$ is a perfect number and odd. At most, $X$ can be divided by all odd numbers until $X/3$: $1…
LocoGris
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Does every perfect number have p*(1) and (p-1)*0 when p is prime at base 2?

For example 6, convert to the base 2 = 110 number of 1's is 2(prime), number of zero's is 1(prime - 1) or 496 = (111110000)2 5(prime) times 1 and 4 times 0 Is this always correct?
Talha ŞAHİN
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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 - (2^n - 2)^2 + (2^n - 1)^2 = P \iff 2^{n - 1}(2^n…
Mr Pie
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Pattern of perfect numbers - why?

I was playing around looking at perfect numbers and noticed some stuff but I don't know much about perfect numbers and would be interested what these patterns amount to. Consider the divisors of some perfect numbers (2 - 7) including themselves…
Curulian
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