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Questions tagged [perfect-numbers]

A perfect number is a positive integer that is equal to the sum of its proper positive divisors.

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Is $N - \varphi(N)$ a square, if $N = q^k m^2$ is an odd perfect number with special prime $q$?

This question was inspired by this MSE question. In MSE, it is shown that $$n - \varphi(n) = (2^{p-1})^2$$ if $n = {2^{p-1}}(2^p - 1)$ is an even perfect number. Here is my question in this post: Is $...
Jose Arnaldo Bebita Dris's user avatar
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If $p^k m^2$ is an odd perfect number with special prime $p$, then $p^k < 2am$ for some positive integer $a < m$

(Preamble: Andy Putman asserts, in the comments, that MO policy prohibits "requests to check completeness of proofs". I have therefore trimmed down my original question to the bare ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
120 views

If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?

My present question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$? It is known that $m^2 - p^k$ is ...
Jose Arnaldo Bebita Dris's user avatar
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If $p^k m^2$ is an odd perfect number with special prime $p$, is it possible to have $p = k$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. My question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, is it ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
1 answer
271 views

On the largest prime factor and the largest component of an odd perfect number

(1) The largest component $p^a$ of an odd perfect number is known to be greater than $10^{62}$. (2) The largest prime of an odd perfect number is known to be greater than $10^8$. Does (1) imply ...
Pascal Ochem's user avatar
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1 answer
403 views

On a GCD approach to odd perfect numbers

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Let $\sigma(z)$ denote the classical sum of divisors of the positive ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
0 answers
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Abelian characters and odd perfect numbers?

This question is about applications of abelian characters to odd perfect numbers: Context and Definitions: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring ...
mathoverflowUser's user avatar
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What can be said about $\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$?

What can be said about the quantity $$\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$? In particular, can one prove that it is always greater ...
Jose Arnaldo Bebita Dris's user avatar
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On "Euclidean" odd perfect numbers

In what follows, we let $N = r^s u^2$ be an odd perfect number given in Eulerian form, i.e. $r$ is the special prime satisfying $r \equiv s \equiv 1 \pmod 4$ and $\gcd(r,u)=1$. In this preprint, ...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
372 views

Steuerwald's theorem

Background: The perfect numbers are the positive integers $n$ such that $$\sigma(n)=2n,$$ where $\sigma(n)$ is the sum of divisors function. The function $\sigma(n)$ is multiplicative and satisfies $$\...
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4 votes
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On Carmichael function and aliquot parts of odd perfect numbers

I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
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Given that $H = \frac{n^2}{\sigma(q^k)/2} = G \times J^2$, where $q^k n^2$ is an odd perfect number, then what is the value of $\gcd(G, J)$?

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
4 votes
1 answer
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Divisibility relation with a specific sum of divisors

Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$ I have checked this up to $n=100$, and I ...
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Alternative proofs of Euclid-Euler theorem

What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
Ross Ure Anderson's user avatar
1 vote
1 answer
144 views

Number of distinct near-squares primes dividing an odd perfect number

I'm curious about if the following question is in the literature or what work can be done about it. Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
user142929's user avatar
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1 answer
196 views

If $p^k m^2$ is an odd perfect number with special prime $p$, then under what other conditions on $\sigma(p^k)/2$ does $k=1$ follow?

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Descartes (1638), Frenicle (1657), and subsequently [Sorli (2003) - ...
Jose Arnaldo Bebita Dris's user avatar
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Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?

Write the odd perfect number $m=p^k a^2$ as a product of primes $$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$ (Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
2 answers
233 views

Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."

I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
jvkloc's user avatar
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
Jose Arnaldo Bebita Dris's user avatar
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0 answers
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On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$

(Preamble: This question is an offshoot of this answer to an MSE question with the same title.) Denote the classical sum of the divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$ and the ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
724 views

Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
319 views

On odd perfect numbers and a GCD - Part III

Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
Jose Arnaldo Bebita Dris's user avatar
8 votes
1 answer
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Can perfect numbers be seen $p$-adically?

It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime. As the very defining property of such a perfect number is to fulfill the ...
Sylvain JULIEN's user avatar
1 vote
1 answer
201 views

On the Diophantine equation $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$

This question is an offshoot of this closely related MO question. Here, we consider the Diophantine equation $$m^2 - p^k = 2^r t,$$ where $r \geq 2$ and $\gcd(2,t)=1$. Furthermore, we place the ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
1 answer
339 views

On odd perfect numbers $q^k n^2$ satisfying $n^2 - q^k = 2^r t$

Let $N = q^k n^2$ be an odd perfect number with special prime $q$, satisfying $$n^2 - q^k = 2^r t$$ where $r \geq 2$ and $\gcd(2,t)=1$. We could prove that: (1) $2^r t > 2n$. (We can modestly ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
2 answers
384 views

Improving the lower bound $I(n^2) > \frac{2(q-1)}{q}$ when $q^k n^2$ 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 $\gcd(q,n)=1$. Define the abundancy index $$I(x)=\frac{\sigma(x)}{x}$$ where $\sigma(x)$ ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
339 views

On odd perfect numbers and a GCD - Part II

(Note: A detailed version of this question was posted in MSE last April 15, 2020. It has not received any responses there as of yet. I have therefore cross-posted it here, hoping that it is ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
2 answers
631 views

On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

This question has been cross-posted from this MSE question and is an offshoot of this other MSE question. (Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
Jose Arnaldo Bebita Dris's user avatar
5 votes
1 answer
617 views

Can an even perfect number be a sum of two cubes?

A similar question was asked before in https://math.stackexchange.com/questions/2727090/even-perfect-number-that-is-also-a-sum-of-two-cubes, but no conclusions were drawn. On the Wikipedia article of ...
player3236's user avatar
2 votes
0 answers
93 views

Geometry for an odd perfect number? (Second question)

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{R})$. Let $h(n) = J_2(n)$ be the second Jordan totient function. Define: $$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
mathoverflowUser's user avatar
1 vote
1 answer
582 views

A geometric approach to the odd perfect number problem?

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$. Let $h(n) = J_2(n)$ be the second Jordan totient function. Define: $$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
mathoverflowUser's user avatar
7 votes
2 answers
574 views

Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

This question is related to the last question about van der Pol's identity for the sum of divisors. In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
mathoverflowUser's user avatar
9 votes
0 answers
677 views

Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
Perfect Number's user avatar
5 votes
0 answers
168 views

Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
Perfect Number's user avatar
4 votes
0 answers
203 views

Frey's elliptic curve and perfect numbers?

Let $E_n:y^2=x(x-\sigma(n)/2)(x+\sigma(n)/n)$ be a Frey-elliptic curve, where $\sigma$ denotes the sum of divisors of the natural number $n$. If $n$ is a perfect number ($\sigma(n)=2n$) then the $j$-...
Perfect Number's user avatar
7 votes
1 answer
389 views

Perfect numbers, Galois groups and a polynomial

Let $f(n,t) = \sum_{k=0}^{r-1} d_k t^k$ where $D_n = \{d_0=1,d_1,\cdots,d_{r-1}\}$ are all divisors of $n$. For instance $$f(28,t) = 28 t^{5} + 14 t^{4} + 7 t^{3} + 4 t^{2} + 2 t + 1$$ For even ...
user avatar
6 votes
1 answer
238 views

Inductively computing Mersenne primes / perfect numbers?

For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$. Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$. Set $\hat{\phi}(1) = \{x_1\}$ and ...
user avatar
4 votes
1 answer
392 views

The action of the unitary divisors group on the set of divisors and odd perfect numbers

Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}\mid d^2 \...
user avatar
2 votes
0 answers
112 views

If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?

My question is as in the title: If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$? I quote from an answer by Varun Vejalla to a closely ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
74 views

Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers

I've cross-posted this from the post of Mathematics Stack Exchange that I've asked (Apr, 2nd 2020) with title On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\...
user142929's user avatar
3 votes
1 answer
371 views

Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes

I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
user142929's user avatar
2 votes
1 answer
192 views

Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
user142929's user avatar
1 vote
1 answer
179 views

Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives

In this post we denote the sum of divisors function as $$\sigma(n)=\sum_{1\leq d\mid n}d,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...
user142929's user avatar
2 votes
2 answers
221 views

A possible axiomatic characterization of the set of divisors of a perfect number

Define a pro-perfect set $S$ to be a finite set of positive integers satisfying the following three properties: $1\in S$. $\displaystyle\sum_{n\in S}n^{-1}\in S$ There exists a unique permutation $\...
Sylvain JULIEN's user avatar
2 votes
0 answers
67 views

Is it possible to deduce statements for odd perfect numbers from the convolution sums involving divisor functions or other arithmetic functions?

Dividing and using some identities of [1] I've deduced the following facts, see also the remarks below. After these introductory paragraphs, to motivate our question, I am asking if we can deduce some ...
user142929's user avatar
1 vote
0 answers
55 views

Equations involving quasiperfect numbers: a first search of odd solutions for this type of equations or well succinct reasonings about these

In this post we study the following equations that involve quasiperfect numbers, denoted as $x$, that are integers such that the sum of all its positive divisors is equals to $2x+1$, and certain ...
user142929's user avatar
2 votes
0 answers
56 views

On $\sum_{\substack{1\leq d\mid n\\d<f(n)}}d$ and odd perfect numbers, for $f(n)$ the greatest prime factor or $\operatorname{rad}(n)$, respectively

First, in this paragraph we remember the definitions/notations for two number theoretic functions, for an integer $m>1$, we denote its greatest prime factor as $\operatorname{gpf}(m)$, and the ...
user142929's user avatar
3 votes
1 answer
375 views

Bounds for the number of prime numbers less than the Euler's factor, the radical and the greatest prime factor, respectively, of an odd perfect number

As tell us the Wikipedia section dedicated to Odd perfect numbers (please, see also the related references if you need it), any perfect number has the form $$n=q^\alpha m^2$$ where the integer $\alpha\...
user142929's user avatar
4 votes
2 answers
599 views

What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
user142929's user avatar
1 vote
0 answers
222 views

Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture

It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
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