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.

1
vote
1answer
102 views

Show that if an odd perfect number exists, it must be divisible by at least 3 different primes

I would assume you'd start by showing that $\dfrac{p}{p-1}\cdot\dfrac{q}{q-1}< 2$ but I don't know how to show that nor how to continue afterwards
2
votes
1answer
269 views

Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?

A positive integer $N$ is said to be $k$-multiperfect if $$\sigma(N) = kN$$ where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer. (The case $k = 2$ reduces to the ...
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 ...
3
votes
0answers
335 views

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 ...
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 ...
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).
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 $825$.
3
votes
2answers
243 views

How are 10-20 digit multiperfect and hemiperfect numbers efficiently computed?

This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just ...
3
votes
2answers
378 views

Prove that if n is an even perfect number then $\sigma(\sigma(n)) < 6n$

$\sigma(n)$ refers to the sum of all divisors function. If n is an even perfect number, then $\sigma(n) = 2n$, but why is $\sigma(\sigma(n)) < 6n$?
1
vote
2answers
176 views

find the largest perfect number less than $10,000$ in Maple

Can anyone tell me how to find largest perfect number less than 10000 in maple? Actually, I know how to find all the perfect numbers less than or equal to 10000 but I don't know how to find the ...
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 $\...
2
votes
2answers
1k views

an odd perfect number cannot be a prime number or a product of two prime numbers or power of prime number. [duplicate]

how to prove : an odd perfect number cannot be a prime number or a product of two prime numbers or power of prime number.
0
votes
1answer
450 views

the last decimal digit of every even perfect number is always 6 or 28. [duplicate]

how to Prove that the last decimal digit of every even perfect number is always $6$ or $28$.
2
votes
4answers
1k views

If no odd perfect numbers exist (or it is unknown that they do) then how can theorems and proofs exist for them?

In my lecture notes, we have been given the theorem If $N \in \mathbb{Z}_{+} \setminus \{1\}$ is odd and perfect, and written $\prod_{i = 1}^k p_i^{n_i}$ as shown, then $k \geq 3$, that is $N$ has ...
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 $(8n +1)$...
0
votes
1answer
247 views

even perfect numbers and primes

Thank in advance to m.s.e site. I am looking for discussions/proof of the 1) If $p ($$2^r$$/2$) is an even perfect number then $p$ should be in the form of $2^r$ - 1 2) Every even perfect 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 ...
8
votes
1answer
780 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.
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 ...
0
votes
1answer
2k views

What is the computational complexity of a brute force perfect numbers finder algorithm?

A loop goes thru all numbers from one to N to find perfect numbers. For each number in the range, it checks all numbers less than it to see if it's a divisor by modding it by the number and checking ...
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 ...
2
votes
3answers
2k views

Perfect numbers, the pattern continues

The well known formula for perfect numbers is $$ P_n=2^{n-1}(2^{n}-1). $$ This formula is obtained by observing some patterns on the sum of the perfect number's divisors. Take for example $496$: $$...
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 $2^p-1$...