Linked Questions
13 questions linked to/from Elementary solution of exponential Diophantine equation $2^x - 3^y = 7$.
2
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1
answer
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Find all integer solutions of $3^a +7 = 2^b$ [duplicate]
I want to find all integer solutions of $3^a + 7 = 2^b$
I have found (by brute force) the two solutions
$3^0 + 7 = 2^3$ and
$3^2 + 7 = 2^4$
but I want to see if there are more solutions. I have ...
- 212
17
votes
6
answers
1k
views
Finding solutions to the diophantine equation $7^a=3^b+100$
Find the positive integer solutions of the diophantine equation $$7^a-3^b=100.$$
So far, I only found this group $7^3-3^5=100$.
- 92.5k
5
votes
1
answer
1k
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Exponential Diophantine equation $7^y + 2 = 3^x$ [closed]
Find all positive integer solutions to $$7^y + 2 = 3^x.$$
ATTENTION: MY SOLUTION HAS A TERRIBLE MISTAKE WHICH I HAVE OVERLOOKED!
Obviously, $x > y$. Then, we have $3^x = 7^y + 2 \equiv 0 \pmod {3^...
user98186
7
votes
2
answers
611
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Solve $13^x-3^y=10$ in positive integers
As it is "well-known" fact $13-3=10$. Is this true for some other powers of $13$ and $3$, i.e. find all natural numbers $x$ and $y$, such that $13^x-3^y=10$ (there are other, find them all).
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4
votes
2
answers
677
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Solve for integer $m,n$: $2^m = 3^n + 5$
Solve for integer $m$ and $n:$ $$2^m = 3^n + 5$$
My Attempt:
Easy to guess two solutions namely $(3,1)$ and $(5,3)$.
Also easy to see that a solution will exist iff $m > 0$ and $n > 0$.
...
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3
votes
2
answers
712
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Find positive integer $x,y$ such that $7^{x}-3^{y}=4$
Find all positive integers $x,y$ such that $7^{x}-3^{y}=4$.
It is the problem I think it can be solve using theory of congruency. But I can't process somebody please help me .
Thank you
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1
vote
4
answers
372
views
For what pair of integers $(a,b)$ is $3^a + 7^b$ a perfect square.
Problem: For what pair of positive integers $(a,b)$ is $3^a + 7^b$ a perfect square.
First obviously $(1,0)$ works since $4$ is a perfect square, $(0,0)$ does not work, and $(0,1)$ does not work, so ...
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2
votes
2
answers
153
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Solve this equation $12^x-5^y=19$ positive integers
Find all $x,y$ be positive integers,such
$$12^x-5^y=19$$
I found $(x,y)=(2,3)$ is solution,maybe have other,so I consider case $x,y>3$ and $\pmod 9$,since
$$12^x\equiv 0\pmod 9,x\ge 2$$
then $5^y\...
- 92.5k
-2
votes
2
answers
245
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Diophantine equation $2^x-3^y=2021$ [closed]
$$2^x-3^y=2021$$
where $x,y$ are non-negative integers. I only found $2^{11}-3^3=2021$.
- 103
2
votes
5
answers
181
views
The diophantine equation $5\times 2^{x-4}=3^y-1$
I have this question: can we deduce directly using the Catalan conjecture that the equation
$$5\times 2^{x-4}-3^y=-1$$
has or no solutions, or I must look for a method to solve it. Thank you.
- 399
-1
votes
1
answer
180
views
Equation in integers $7^x-3^y=4$
I don't know how to solve $7^x-3^y=4$...
I tried to see something $\pmod 7$ and $\pmod 3$ but it doesn't help at all. Can anyone give me some hints about it?
2
votes
3
answers
138
views
Find all integer numbers $k$,$l$,$m$,$n$ for which $3^k$ • $5^l$ + $7^m$ = $2^n$
Find all integer numbers $k$,$l$,$m$,$n$ for which $3^k\cdot 5^l+7^m=2^n$.
I tried to find a decent modulo and look at ending digits, but nothing more. Thank you for your responses.
Obviously is $k$...
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8
votes
1
answer
277
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Solutions for diophantine equation $3^a+n=2^b$
A few days ago I asked for a solution for $3^a+1=2^b$ where $a\in \Bbb N$ and $b\in \Bbb N$ (Solutions for diophantine equation $3^a+1=2^b$), and I got two good answers for this question.
But there I ...
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