Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
35 views

prove that $n - \phi(n)$ is a square where $n$ is an even perfect number

where even perfect numbers are of the form $2^{p-1}(2^{p} - 1)$ ( $p$ and $2^{p} - 1$ are prime numbers ) My attempt $\phi(n)$ = ($2^{p - 1} - 2$)($2^{p} - 2$) So, we need to prove that. $2^{p - 1}$($...
Oppenheimer's user avatar
1 vote
1 answer
46 views

Distribution of perfect numbers for a semiprime

Given a semiprime with a length of 120 digits (397bit): is it possible to meet any assumptions about perfect numbers (prime factors with same length, 199+199bit) for this number? I have made an ...
Alex Tbk's user avatar
  • 121
4 votes
0 answers
57 views

Can we efficiently check whether a number is a Zumkeller number?

A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum. If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
Peter's user avatar
  • 84.8k
-1 votes
1 answer
165 views

Can we really be sure that there is no odd perfect number below $10^{3000}$?

A positive integer $N$ is called perfect if the sum of its divisors (including $1$ and $N$) is $2N$. A famous open problem is whether there is an odd perfect number. Can someone confirm the following ...
Peter's user avatar
  • 84.8k
1 vote
0 answers
52 views

patterns in the abundancy index of integers

Let $\sigma(n)$ be the sum of all divisors (including 1 and $n$) of $n$, and define the abundancy index of $n$ as $I(n) = \sigma(n)/n $. For example: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. ...
AndroidBeginner's user avatar
2 votes
1 answer
50 views

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 ^ ...
louis's user avatar
  • 35
2 votes
0 answers
96 views

Can a Odd Perfect Number be a Carmichael Number?

Can a odd perfect number $n$ be a Carmichael number? We know that all Carmichael numbers are odd and square-free. But is there a Carmichael number that is also a perfect number? We all know that if ...
Thirdy Yabata's user avatar
1 vote
0 answers
70 views

Is there any square harmonic divisor number greater than $1$?

A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, $n$ is a harmonic divisor number if and only if $\dfrac{nd(n)}{\sigma(n)...
Jianing Song's user avatar
  • 1,943
0 votes
2 answers
64 views

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 $...
IONELA BUCIU's user avatar
  • 1,115
0 votes
1 answer
96 views

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 ...
Cain Goldhardt's user avatar
1 vote
1 answer
209 views

Is $28$ the only perfect number that is form of $n^n+1$?

Is $28$ the only perfect number that is form of $n^n+1$? I noticed that in the sequence of $n^n+1$, $28$ is the only perfect number that is in the list. Using Pari GP, I checked the values of $n\leq10^...
Thirdy Yabata's user avatar
0 votes
1 answer
49 views

Specific Prime Numbers Chain proofs of maximum length limits.

Considering Prime Numbers $p\in P$ such that $(p^2+4)\in P, (p^2+4)^2+4\in P, ((p^2+4)^2+4)^2+4\in P, (((p^2+4)^2+4)^2+4)^2+4\in P,((((p^2+4)^2+4)^2+4)^2+4)^2+4\in P$, seems to be either very rare, or ...
Eugen's user avatar
  • 238
0 votes
0 answers
40 views

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 ...
blademan9999's user avatar
1 vote
1 answer
297 views

Who discovered the largest known $3$-perfect number in $1643$?

Multiperfect numbers probably need no introduction. (These numbers are defined in Wikipedia and MathWorld.) I need the answer to the following question as additional context for a research article ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
393 views

Under what conditions is it true that $m^2 - a$ is not a square if and only if $(m - 1)^2 < m^2 - a < m^2$, where $a>0$?

Preamble: The present inquiry is an offshoot of this MSE question from February 21, 2019, and ultimately, Theorem III.2, page 2 from a paper submitted to a conference organized by De La Salle ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
3 answers
373 views

Odd perfect numbers

In this other question, somebody mentions that in a letter to Mersenne dated November 15, 1638, Descartes showed that $D=3^2⋅7^2⋅11^2⋅13^2⋅22021=198585576189$ would be an odd perfect number if $22021$ ...
Dominique's user avatar
  • 2,391
2 votes
2 answers
182 views

Does an odd perfect number have a divisor (other than $1$) which must necessarily be almost perfect?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. Denote the aliquot sum of $x$ by $s(x)=\sigma(x)-x$ and the deficiency of $x$ by $d(x)=2x-\sigma(x)$. ...
Jose Arnaldo Bebita Dris's user avatar
-1 votes
1 answer
252 views

Can a repunit number be a perfect number? [closed]

Can a repunit number be a perfect number? I think a repunit number can be a perfect number, since the last digit of an odd perfect number can be $1$. Am I correct?
Thirdy Yabata's user avatar
1 vote
0 answers
140 views

Proving $n \mid \sigma(n^2)$

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
Jose Arnaldo Bebita Dris's user avatar
0 votes
0 answers
41 views

Is this proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, where $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs $$G = \gcd(\sigma(q^k),\...
Jose Arnaldo Bebita Dris's user avatar
-1 votes
1 answer
205 views

Can $1!+2!+3!+\cdots n!$ be a perfect number? [closed]

Can $\sum_{k=1}^nk!$ be a perfect number? I think an odd perfect number can be divisible by $9$ and the last digit of an odd perfect number can be $3$, so I think the above sum can be a perfect number....
Thirdy Yabata's user avatar
4 votes
1 answer
276 views

Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$

Preamble: The present inquiry is an offshoot of What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?. MOTIVATION Denote the classical sum of ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
120 views

Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. It is known that $$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$ where $D(x)=2x-\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
129 views

If $p^k m^2$ is an odd perfect number, then $D(p^k)/s(p^k)$ is in lowest terms. Does this contradict $D(p^k)D(m^2)=2s(p^k)s(m^2)$?

In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect number. Euler showed ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
136 views

Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer. The number $1$ ...
Jose Arnaldo Bebita Dris's user avatar
-4 votes
1 answer
291 views

Are there any perfect numbers other than $28$ that are of the form of $\,n^3+1\,?$ [closed]

I am wondering if there is any perfect number that is of the form $\,n^3+1$ (i.e. one more than a perfect cube). Since perfect number minus one (for example, $495$ and $8127$) $\equiv$ $0\pmod{9}$ and ...
Thirdy Yabata's user avatar
1 vote
1 answer
137 views

If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.

While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove: CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
67 views

On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
Jose Arnaldo Bebita Dris's user avatar
0 votes
2 answers
119 views

If $q^k n^2$ is an odd perfect number, then $n^2 - q^k = 2^r t$ implies that $3 \leq r$ is odd. Therefore?

The topic of odd perfect numbers likely needs no introduction. Let $N$ be an odd perfect number given in the so-called Eulerian form $N = q^k n^2$ where $q$ is the special prime satisfying $q \equiv k ...
Jose Arnaldo Bebita Dris's user avatar
-1 votes
1 answer
97 views

On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)

Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
195 views

Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II

This MSE question (from April 2020) asked whether the inequality $$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$ could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
131 views

On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$

Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2$$ where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, let ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
160 views

On Tony Kuria Kimani's recent preprint in ResearchGate

(Preamble: The method presented here to compute the GCD $g$ is patterned after the method used to compute a similar GCD in this answer to a closely related MSE question.) Let $\sigma(x)=\sigma_1(x)$ ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
1 answer
169 views

Is $P+1$ prime for the perfect number $P$ corresponding to the exponent $74207281$?

The even perfect numbers are closely related to the Mersenne primes. We currently know $51$ Mersenne primes and hence $51$ perfect numbers. It has already been checked for which of those perfect ...
Peter's user avatar
  • 84.8k
0 votes
1 answer
96 views

Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2))=D(n^2)/s(q^k)$ - Part II

(Preamble: This inquiry is an offshoot of this answer to a closely related question.) In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
70 views

Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II

(Preamble: This inquiry is an offshoot of this MSE question.) Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
67 views

Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$. It is known that $$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
169 views

Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2)) = D(n^2)/s(q^k)$

(Preamble: This question is an offshoot of this earlier post.) In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
117 views

Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?

In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
Jose Arnaldo Bebita Dris's user avatar
0 votes
2 answers
115 views

Is there any other known relationship between even perfect numbers and odd perfect numbers, apart from their multiplicative forms?

(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.) Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
87 views

Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?

The topic of odd perfect numbers likely needs no introduction. Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
119 views

Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?

Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
76 views

Follow-up to MSE question 3738458

This is a follow-up inquiry to this MSE question. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
Jose Arnaldo Bebita Dris's user avatar
0 votes
0 answers
69 views

Primes with prime digits and prime sum of the digits

I don't know if this concept is already defined, I consider the number $n$ to be a perfect prime (weak) if $n$ is a prime, its digits are primes and the sum of the digits is prime. Every one digit ...
Alex's user avatar
  • 1
0 votes
0 answers
43 views

Can this inequality be solved for $q$ in terms of $n$ (or the other way around), if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. The following general inequality relating $k$ and $q$ is proved in this ...
Jose Arnaldo Bebita Dris's user avatar
0 votes
1 answer
101 views

What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
93 views

On Carmichael function and aliquot parts of odd perfect numbers

This post is cross-posted on MathOverflow with identifier 439563 and same title. We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\...
user759001's user avatar
0 votes
0 answers
94 views

Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs: $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \...
Jose Arnaldo Bebita Dris's user avatar
4 votes
1 answer
82 views

Reference request regarding odd 4-perfect numbers

I am reading a paper by Broughan and Zhou (2006) that is dealing with odd 4-perfect numbers. The title of the paper is "Odd multiperfect numbers of abundancy four." In this paper they ...
User4576283's user avatar
4 votes
1 answer
197 views

semi perfect number - number of divisors

The definition: we define number as semi perfect , if the number equals to the sum of exactly k of its divisors. the question: prove that for every n (n>0 | n belong to N) n is semi perfect order 3 ...
elad's user avatar
  • 55

1
2 3 4 5
10