# Questions tagged [exponential-diophantine-equations]

Diophantine equation where the variable is in the exponent (ex.: find all sols for $3^x-5^y=7$)

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### Solutions to $4^\alpha+1=q^\beta$ [duplicate]

Problem. Solve the equation $$4^\alpha+1=q^\beta$$ for prime number $q$ and positive integers $\alpha,\beta$. Background. This question arose in the context of the following question: Find all ...
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### What does it take to find integer solutions to this exponential division equation?

Consider this equation where $a$ and $b$ are positive integers. $$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$ This equation has the trivial solution $k=1, a=1, b=1$. How would I find more solutions, or show ...
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### How to prove there exists at least one positive integer solution of $k$ and $n$ in $2^{k} = 3^{n-1}(m + 1) - 1$, where m is any even integer?

Here's the related question Is it possible to prove there exists at least one positive integer solution for $k$ and $n$ in $2^{k} = 3^{n-1}(m + 1) - 1$, where m is any even positive integer? So we ...
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### Understanding the Lambert W function

I am asking this question because I wish to get a feel for how it would be to invent the Lambert W function. I would like to really understand it. We have the equation: $$xe^x=y \tag{1}$$ and we want ...
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### The solutions of the diophantine equation $z^x=1+(z-1)y^2$

I'm trying to determine the positive integer solutions of the diophantine equation $z^x=1+(z-1)y^2$. This equation can be written also as $$y^2=\frac{z^x-1}{z-1}=\sum_{i=0}^{x-1}z^i$$ Trivial ...
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### The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$,$y$ are integers

The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$,$y$ are integers I tried using hit and trial and got only one solution $x=3$ and $y=1$ which is the answer. How can we prove that it is the ...
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### Collatz Conjecture: For a cycle where the maximum odd integer is $x_{max}$, does it follow that $x_{max} < 3^n$

I am working on understanding the upper limit in the case where a non-trivial cycle exists for the Collatz Conjecture. Is the following reasoning valid for establishing that the maximum odd integer in ...
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### Establishing a lower bound with integers $m,n$ for $2^m - 3^n$ when $2^m > 3^n$

When I look at the encyclopedia of integers, it is clear that this grows even though it doesn't grow continuously. It seems to me that it should be possible to establish a lower bound based on ...
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1 vote
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### For integers $m>2, n>1, c> 0$, with $2^m - 3^n=c$, does it follow for integers $x>0,y>0$ that $2^{m+x} - 3^{n+y} \ne c$

I am attempting to generalize an answer given by Will Jagy here. I may have made a mistake since the conclusion is stronger than I typically have seen and my argument is pretty much the same as Will's....
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