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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.

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231 views

Prove that if $2^a$ is superperfect, then $2^{a+1} - 1$ is a Mersenne prime.

Definition: Let $n \in \Bbb Z$ with $n>0$. Then $n$ is said to be superperfect if $σ(σ(n))=2n$. Where $σ$ is the sum of positive divisors arithmetic function. I am trying to solve a proof that ...
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53 views

There are at most finitely many primes not dividing any elements in this sequence

A perfect number is a natural number $n$ that is the sum of its proper divisors. Otherwise said, they are the natural numbers $n$ such that $\sum_{d|n}d=2n.$ For instance, $6$ and $28$ are perfect. ...
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278 views

Why are non-trivial powers called perfect powers?

Let number mean positive integer. What makes a number perfect is quite intricate: A perfect number is a number that is equal to the sum of its proper divisors. What makes a number a power is also ...
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151 views

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 ...
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168 views

Explain these concepts of String Theory in easy words for mathematicians, from a popular point of view

In the Wikipedia's article for the number 496 in the section Physics is related the condition found by Green and Schwarz about this perfect number in their string theory. Question. Can you tell us ...
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174 views

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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200 views

On the undecidability of the existence of odd perfect numbers

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. That is, $$\sigma(x) = \sum_{d \hspace{0.01in} \mid \hspace{0.01in} x}{d}.$$ For example, the divisors of $28$ are $$1, 2, ...
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59 views

If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, does the following implication hold?

Let $\sigma(x)$ denote the sum of the divisors of $x$. For example, $\sigma(6)=1+2+3+6=12$. Numbers $N$ satisfying the equation $\sigma(N)=2N$ are called perfect numbers. Euler showed that an odd ...
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99 views

On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

(Note: This question has been cross-posted to MO.) Let $\sigma$ be the classical sum-of-divisors function. For example, $$\sigma(6)=1+2+3+6=12={2}\cdot{6}.$$ If $\sigma(N)=2N$ and $N$ is odd, then $...
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55 views

On identities and congruences involving the harmonic mean of odd perfect numbers

I would like to know if my calculations were rights and if you want to deduce other different interesting congruences following my approach. On assumption (notice that it isn't the usually accepted ...
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51 views

Is the following statement true if $N = q^k n^2$ is an odd perfect number given in Eulerian form?

A number $N$ is said to be perfect when $\sigma(N)=2N$, where $\sigma$ is the classical sum-of-divisors function. An odd perfect number $N = {q^k}{n^2}$ is said to be given in Eulerian form if $q$ is ...
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45 views

the sum of eight three digit consequtive even numbers is S.When Sis divided by 5,it results in a perfect cube

how many sets of such eight numbers are possible? 1,2,3,4,5 (choose among this) which of the following can be one of those eight numbers 644, 328, 108, 126, 140
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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 $n = 3, 1 + 2 + 4 = 7 = 8 - 1$.
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If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 - \frac{5}{3q}$?

Let $\sigma=\sigma_{1}$ be the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q$ satisfies $q \equiv k \equiv 1 \pmod 4$), and $k=1$, does ...
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1answer
59 views

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Euler's theorem for odd perfect numbers states that if there exists and odd perfect number, that is an odd positive integer $n$ satisfying $\sigma(n)=2n$, where $\sigma(m)=\sum_{d\mid m}d$ denotes the ...
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If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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79 views

Can an odd perfect number be divisible by either $2049$ or $2051$?

Can an odd perfect number be divisible by either $2049$ or $2051$? Note that $2049 = 3 \cdot {683}$, and that $2051 = 7 \cdot {293}$. Added July 15 2016 It is known that an odd perfect number ...
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1answer
35 views

If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$?

The title says it all. If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$? Here $I(x)$ is defined to be the ratio $$I(x) = \dfrac{...
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1answer
106 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers

(Note: This question has been cross-posted from MO.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M ...
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287 views

Origin of Almost Perfect Numbers

Let $N$ be a positive integer. $N$ is called a perfect number if the sum of its positive divisors denoted by $\sigma(N)=2N$. For example $6$ is a perfect number since: $\sigma(N)=1+2+3+6=12=2(6)$. ...
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3answers
109 views

Elementary number theory sum of divisors

Let the sum of the divisors of a number $N$ be equal to $s$(excluding N itself) then show that if $s=N$ then show that N is a perfect number. I tried to use the basic formula for sum of divisors but ...
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1answer
38 views

Can $\sigma(2^r)$ be abundant for $r > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(y) < 2y$, $y$ is called deficient; if $\sigma(z) > 2z$, $z$ is called abundant. Questions (1) Can $\...
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perfect powers and perfect numbers

The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect powers and perfect numbers . However, this must be very rare. Is it unique ? Questions: $\;$1) Are there any ...
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1answer
126 views

On the Density of Deficient Odd Numbers and Abundant Integers

Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) ...
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51 views

A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) $$1^2+2^2+\ldots+n^2=\frac{n(n+1)(...
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70 views

For which $r$ the expression $\sqrt{4s+ r^2}$ is an integer? [closed]

Let's assume $s$ is a non-negative composite integer. We need to find such a $r$ (also integer, non-negative and $r < s-1 $), for which the square root $\sqrt{4s + r^2}$ will yield an integer? Any ...
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1answer
56 views

Perfect numbers of the form $12m+1$ and $\sum_{d\mid n}\frac{1}{\phi(d)}$, where $\phi(m)$ is Euler's totient function

If there are no mistakes combining Exercise 9 a) (Chapter 3, page 71) and Exercise (Chapter2, page 47) of Apostol's Introduction to Analytic Number Theory we can prove easily Lemma. If $n$ is a ...
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1answer
72 views

Even perfect numbers and a relationship with polygonal numbers

Let $\sigma(m)=\sum_{d\mid m}d$ the sum of divisors function, for example $\sigma(6)=1+2+3+6=12$. Question. I don't know if this exercise was in the literature, and I believe that I know how prove ...
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88 views

Odd Perfect Numbers attempt

Suppose you have some positive, odd, composite integer $n$. Let $$ D = d_1 + d_2 +d_3+...+d_{j-1} $$ where $d_i$ is some divisor of $n$ and $j$ is an odd number corresponding to the number of proper ...
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2answers
61 views

Proving this quotient is always even.

Let $n$ be an odd, composite integer. Let $D$ equals the sum of the proper divisors of $n$ minus the last divisor, $d_j$. Also, let $n$ be a number such that $D \gt \frac{1}{2} \times{n}$. Let $d_i$ ...
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110 views

Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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4k views

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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29 views

Requesting for details of Grun's result on odd perfect numbers

This post is an offshoot of this earlier MSE question. According to this Wikipedia page: Grun $1952$ proved that the smallest prime factor $p_1$ (of an odd perfect number $N$) is $< \frac{2\...
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94 views

Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...
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65 views

If $b$ is an odd composite number and $\dfrac{b^2 - 1}{\sigma(b^2) - b^2} = q$ is a prime number, what happens when $q = 2^{r + 1} - 1$?

(Note: An improved version of this question has been cross-posted to MO.) Let $\sigma(X)$ be the sum of the divisors of $X$. For example, $\sigma(2) = 1 + 2 = 3$, and $\sigma(4) = 1 + 2 + 4 = 7$. ...
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1answer
43 views

Why $\frac{2a+2b-4}{5b-4}<1$?

I was reading this paper. I didn't understand something on the 5th page: "We may suppose that $a\geq b>0$. If $b\geq 5$ then $\frac{2a+2b-4}{ab-4}\leq \frac{2a+2b-4}{5b-4}<1$." Why? $\frac{2a+...
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1answer
28 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, does $\sigma(n^2)/q$ divide $2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ be the sum of the divisors of $x$. A number $X$ is called perfect if $\sigma(X) = 2X$. Denote the abundancy index $\sigma(X)/X$ by $I(X)$. If $N$ is odd and perfect, then $N$ can be ...
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1answer
51 views

Can the existence of infinitely many even perfect numbers be settled by a diagonal argument?

Say a (finite or not) sequence of strictly increasing positive integers $(u_{i})_{i\in I}$ is a 'Euclid sequence' if and only if the sum of reciprocals of all the $(u_i)$ equals $2$. Now suppose we ...
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1answer
247 views

Why did Euclid call 6 a perfect number?

The old Greek did not consider $1$ a number. Nevertheless Euclid called $6 = 1+2+3$ a perfect number. How could he use $1$ which was not a number?
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73 views

Mersenne primes: All even and perfect number has the form $2^{p-1}M_p$ for some prime $p$, being $M_p$ a prime of Mersenne.

All even and perfect number has the form $2^{p-1}M_p$ for some prime $p$, being $M_p$ a prime of Mersenne. What I did was: Suppose that $n = 2^kb$ and $\sigma(n) = 2n$. Then, $\sigma(2^kb) = (2^{k+1}...
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60 views

Reference request: Do any papers on odd perfect numbers approach the problem using the following equation?

(Note: This question has been cross-posted to MO.) Do any papers on odd perfect numbers approach the problem using the following equation? $$N - (q^k + n^2) + 1 = \sigma(q^{k-1})(q-1)(n+1)(n-1)$$ ...
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150 views

If n>6 is an even perfect number, Prove that n is congruent to 4 (mod12)

I know that for n to be an even perfect number greater than 6 it has the form $ 2^(m-1)(2^m-1)$ where m is prime. I also know that since n is an even perfect number, it is congruent to 1 (mod 9). ...
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48 views

On the number of $n$-perfect numbers

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. (e.g. $6$ is a perfect number) ...
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1answer
93 views

If $n$ satisfies $\left(-3+\sqrt{1+8n}\right)\sigma(n)=4\left(-1+\sqrt{1+8n}\right)\phi(n)$ then is an even perfect number?

Let an integer $m\geq 1$, and $\sigma(m)$ is the sum of positive divisors function, and $\phi(m)$ is Euler's totient function, counting the number of integers $1\leq k\leq m$ such that $gcd(k,m)=1$ (...
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55 views

With $rad(N)=\prod_{p|N}p$, if $N$ is even and $\frac{2+rad(N)}{8}\left(\sum_{\substack{d|N,d<rad(N)}}d\right)=N$ then is perfect?

In the literature (see for example sites and paper concerning to the abc conjecture, I say this as reference and by caution to avoid mistakes) is defined the arithmetical function $rad(n)$ as $1$ if $...
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1answer
58 views

Iterated $\sigma^k(n)$, excluding the possibility that primes $p|\sigma^t(n)$, $t< k$ divides an odd perfect number $n$

Let $m\geq 1$ an integer and $\sigma(m)=\sum_{d|m}$ the sum of positive divisors function. A positive integer is said to be perfect if and only if $\sigma(n)=2n$. We have that $\sigma(m)$ is a ...
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1answer
99 views

Eulers Phi function with list of distinct perfect numbers

Prove that $\phi(n_1 n_2 \cdots n_m)$ = $2^{m-1}\cdot \phi(n_1)\cdot\phi(n_2) \cdots \phi(n_m)$ where all of the $n$'s are distinct even perfect numbers. I thought that because $\phi(2^k) = 2^{k-1}$ ...
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1answer
68 views

Looking for one of B. Hornfeck's papers on odd perfect numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A number $N$ is called perfect if $\sigma(N)=2N$. According to this ...
4
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1answer
106 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...
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1answer
24 views

What doesn 2nd column depict in this table for perfect number description?

I am reading about perfect number at below link. http://mathworld.wolfram.com/PerfectNumber.html I am not able to understand what does the 2nd column (pn) depict? ...