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Results tagged with nt.number-theory
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user 10365
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
0
votes
For $q$ prime and $\gcd(q, n) = 1$, can $N = qn^2$ be almost perfect?
My bad, I just realized the following: If $N$ is an odd almost perfect number, then since $\sigma(N) = 2N - 1$ is also odd, then $N$ has to be a square.
Consequently, for my question that:
…
0
votes
Perfect numbers $n$ such that $2^k(n+1)$ is also perfect
[After typing out this attempt at a "partial" answer, I realized that the details have already been worked out by Luis, Gerhard and Todd. I am posting it as an answer for anybody else who might be in …
-1
votes
1
answer
104
views
For $q$ prime and $\gcd(q, n) = 1$, can $N = qn^2$ be almost perfect?
In general, for $q$ prime and $\gcd(q, n) = 1$, can $N = qn^2$ be almost perfect?
An almost perfect number $N$ is a positive integer satisfying $\sigma(N) = 2N - 1$, where $\sigma(x)$ is the sum of a …
1
vote
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd p...
This is too long to be posted as a comment, as I just wanted to share some of my recent thoughts on why it is difficult to obtain such an integer $x \nmid N$, where $N = {q^k}{n^2}$ is a (hypothetical …
2
votes
1
answer
168
views
Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2...
(This post is an offshoot of this MSE question.)
Let $\sigma(x)$ denote the sum of divisors of $x$. (https://oeis.org/A000203)
QUESTION
Is the asymptotic density of positive integers $n$ satisfying …
1
vote
Can $k$ be arbitrarily large in the following equations?
Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$.
Unconditionally, it is known that $q^k < n^2$ [Dris, 2012]. This implies that $k$ and $n$ are dependent, which means that the proof i …
1
vote
2
answers
199
views
What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(...
Let $\sigma(x)$ denote the classical sum-of-divisors function, and let
$$I(x) = \frac{\sigma(x)}{x}$$
be the abundancy index of the positive integer $x$.
My question is this: What proportion of th …
0
votes
1
answer
952
views
Perfect Numbers - On Mersenne and Euler Primes
Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:
In the same way that the Me …
0
votes
Accepted
Perfect Numbers - On Mersenne and Euler Primes
I tried to get in touch with Dean Hickerson and here's what he got to say regarding the problem of determining the status of squares with respect to solitude or friendliness:
Notice that all of th …
2
votes
How do we recognize an integer inside the rationals?
The following paper by Stan Wagon and Dan Flath might be of some interest to you:
How to pick out the integers in the rationals: An application of number theory to logic, American Mathematical Monthl …
9
votes
2
answers
759
views
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd p...
Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
I have asked the same question in MSE, but did not get any answers. I was wondering if anyb …
3
votes
Generalized quasi-perfect numbers
The related equation
$$\sigma(n) = An + B(n)$$
where $B(n)$ "is a function that may depend on properties of $n$" is considered in the paper Variations on Euclid’s Formula for Perfect Numbers by Faride …
0
votes
1
answer
746
views
Question Re: Arian Berdellima's Papers On Odd Perfect Numbers [closed]
Hi everyone.
I'd like to refer you to two papers by Arian Berdellima on odd perfect numbers:
More Properties About Odd Perfect Numbers
http://mpra.ub.uni-muenchen.de/31587/1/MPRA_paper_31587.pdf
P …
5
votes
Algebraic Attacks on the Odd Perfect Number Problem
I agree with Pace - the correct function to consider would be the abundancy index instead of the sigma function itself. In a certain sense, the abundancy index value of 2 for perfect numbers (odd or …
4
votes
2
answers
470
views
On the natural density of almost perfect numbers
This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE.
Essentially, by work of Kanold, we know that the …