0
votes
1answer
69 views

Is there any easy way to find the positive integer solutions $(x,y,z)$ from this linear equation?

The equation is like this: $3^x -2^y = 19^z$ It seems that no way to find the solution except using trial and error. I got only one solution: $x=3, y=3, and z= 1$ by using trial and error. But, ...
2
votes
2answers
124 views

When is the power of a binomial equal to the sum of like powers of its terms?

Question: Under what circumstances/restrictions on $x$ and $y$ does $(x + y)^n = x^n + y^n$ given the value of $n$? That is, what can we tell about $x$ and $y$ from the value of $n$ and the equation ...
8
votes
5answers
143 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
0
votes
5answers
344 views

Generate solutions of Quadratic Diophantine Equation

Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation Here's the answer: $$ ...
3
votes
3answers
766 views

How to solve Quadratic Diophantine Equation

Here's the problem. Find the solutions of the following equation: $$ k^2 - 1 = 5(m^2 - 1).$$ Here's my idea: The original equation can be written as: $$ k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ...
3
votes
1answer
157 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...
8
votes
1answer
469 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
5
votes
2answers
364 views

How many positive integer solutions to $a^x+b^x+c^x=abc$?

How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to ...
40
votes
6answers
3k views

$x^y = y^x$ for integers $x$ and $y$

We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?