All Questions
10
questions
12
votes
3
answers
462
views
On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two
I don't know if the following diophantine equation (problem) is in the literature. We consider the diophantine equation $$x^{m-1}(x+1)=y^{n-1}(y+1)\tag{1}$$ over integers $x\geq 2$ and $y\geq 2$ with $...
6
votes
1
answer
252
views
Consider the following Diophantine equation: $x^2 + xy + y^2 = n$ [duplicate]
Consider the following Diophantine equation
$$x^2 + xy + y^2 = n\,.$$
For a particular positive integer $n$, the number of solutions $\left(x,y\right)$ such that $x$ and $y$ are integers is given by ...
- 443
1
vote
2
answers
577
views
What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$
I saw this in quora.
What are all the
integer solutions to
$a^{b^2} = b^a$
with $a, b \ge 2$?
Solutions I have found so far:
$a = 2^4 = 16, b = 2,
a^{b^2}
= 2^{4\cdot 4}
=2^{16},
b^a = 2^{16}
$.
$...
- 108k
1
vote
0
answers
43
views
Non-Linear Diophantine Equation in Two Variables [duplicate]
How many solutions are there in $\mathbb{N}\times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}$ ? I could solve till I got to the point where $1995^2$ is equal to the ...
- 344
1
vote
1
answer
112
views
Finish the exercise to find or count the solutions of the equation $n x^2-\operatorname{rad}(n)=y^3$ over positive integers
I would like to solve if it is possible next diophantine equation for positive integers $n\geq 1$, $x\geq 1$ and $y\geq 1$
$$n x^2-\operatorname{rad}(n)=y^3,\tag{1}$$
where $\operatorname{rad}(n)$ ...
user243301
0
votes
1
answer
195
views
Finishing the task to find the solutions of $\frac{1}{x}-\frac{1}{y}=\frac{1}{\varphi(xy)},$ where $\varphi(n)$ denotes the Euler's totient function
In this post I evoke a variant of the equations showed in section D28 A reciprocal diophantine equation from [1], using particular values of the Euler's totient function $\varphi(n)$. I ask it from a ...
user243301
0
votes
3
answers
167
views
About the solutions of $x^{\varphi(yz)}+y^{\varphi(xz)}=z^{\varphi(xy)}$, being $\varphi(n)$ the Euler's totient
In this post we denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$. I wondered about the solutions of the equation
$$x^{\varphi(yz)}+y^{\varphi(xz)}=z^{\varphi(xy)}\tag{1}$$
...
user243301
4
votes
1
answer
331
views
Markov triples that survive Euler's totient function
I'm inspired in a recent post of this MSE. We denote the Euler's totient function in this post as $$\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right).$$
Suppose we have three positive integers $a,...
user243301
2
votes
3
answers
443
views
Mathematics Number Theory Proof: Prove that there are no integers $a$ and $b$ such that: $a^2-3b^2 = -1$
Prove that there are no integers $a$ and $b$ such that $a^2-3b^2 = -1$.
I got a hint to use prime factorization, so I rearrange the equation to be $a^2 = 3b^2-1$ and set $a$ to be $a= p_1p_2\cdots ...
- 235
1
vote
2
answers
124
views
The Diophantine Equation $m(n-2016)=n^{2016}$
How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number?
HINT.-There are lots of solutions
HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
- 30.3k