All Questions
Tagged with divisor-sum diophantine-equations
12
questions
13
votes
1
answer
526
views
Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
(Note: This question has been cross-posted to MO.)
Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$.
Here is my question:
Does the equation $\sigma(\sigma(x^2))=2x\...
- 8,010
2
votes
1
answer
46
views
For which primes $p$ and positive integers $k$ is the deficiency $D(p^k)$ equal to the arithmetic derivative of $p^k$?
The Problem
For which primes $p$ and positive integers $k$ is the deficiency $D(p^k)$ equal to the arithmetic derivative of $p^k$?
My Attempt
Let $\sigma(x)$ denote the sum of divisors of the ...
- 8,010
2
votes
1
answer
104
views
Is it possible to derive $m < p^k$ from the Diophantine equation $m^2 - p^k = 4z$ unconditionally, when it is solvable?
This question is an offshoot of this earlier one.
Allow me to state my question in full:
Is it possible to derive $m < p^k$ from the Diophantine equation $m^2 - p^k = 4z$ unconditionally, where ...
- 8,010
1
vote
2
answers
151
views
On the Diophantine equation $m^2 - p^k = 4z$, where $z \in \mathbb{N}$ and $p$ is a prime satisfying $p \equiv k \equiv 1 \pmod 4$
The title says it all:
Do there always exist $m, p, k \in \mathbb{N}$ such that $m^2 - p^k = 4z$, $z \in \mathbb{N}$ and $p$ is prime satisfying $p \equiv k \equiv 1 \pmod 4$?
MY ATTEMPT FOR $z=1$
...
- 8,010
1
vote
0
answers
98
views
Prove that $\sum_{u = 1}^{N} \sum_{v=1}^{N} \left(\left[\sqrt{u^2-4v} \in Z\right]+\left[\sqrt{u^2+4v} \in Z\right]\right) = D \left(N\right)-1$
For a given height $N$. I am performing various sums involved in the counting of the number of monic quadratics, cubics, etc. that factor over a range. I noticed by experiment that the above sum ...
- 967
2
votes
1
answer
102
views
If $q^k n^2$ is an odd perfect number with special prime $q$, then can $n^2 - q^k$ be a cube?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$.
This question is an offshoot of the following earlier post:
If $q^k n^2$ is an odd perfect ...
- 8,010
2
votes
1
answer
239
views
About the equation $x^2+y^2+z^2+2t^2=n$
The question
The final goal (for this stage of my project) is to get an explicit form for $\phi(n)$. This last one is the number of integer solutions to $x^2+y^2+z^2+2t^2=n$. You may find this $\phi(n)...
- 3,813
1
vote
0
answers
120
views
What are the properties of abundancy numbers?
Define abundancy numbers as the rational numbers that are equal to the abundancy index of some integer (not to be confused with «abundant numbers», which are natural numbers with abundancyindex ...
- 15.7k
0
votes
2
answers
79
views
$\sum\limits_{\mathbb{d|n}}{f(d)}=\sum\limits_{\mathbb{d|n}}{g(d)}\implies f(n)=g(n)?$
Question:
Is it true that if for functions $f,g$ which map naturals to naturals
For all natural numbers n, we have $f(n)=g(n) \iff$ for all natural
numbers n we have
$\sum\limits_{\mathbb{d|...
- 3,813
5
votes
2
answers
166
views
Can $\sigma(p^k)=2^n$ for some $k>1$?
If $k=1$ then the only solutions to
$$
\sigma(p^k)=2^n
$$
are when $p$ is a Mersenne prime (of course $p$ is restricted to the primes). Is there a solution for larger $k$? It doesn't seem so but I can'...
- 32.3k
0
votes
1
answer
92
views
When does $\sigma_{1}(p^{2k}) = q$ have a solution, where $p$ and $q$ are primes, and $k \geq 1$?
Let $\sigma_{1}$ be the classical sum-of-divisors function. For example, $\sigma(12)=1+2+3+4+6+12=28$.
Here is my question:
When does $\sigma_{1}(p^{2k}) = q$ have a solution, where $p$ and $q$ ...
- 8,010
3
votes
0
answers
65
views
On Solvability of an Equation with the Sigma Function
I have researched this question and need help. In general is there only one solution to the equation $\sigma(nx) = (n+4)x$ for every $n$? Specifically, is there only one solution to $\sigma(5x) = 9x$...
- 115