Find all solutions to the diophantine equation $7^x=3^y+4$ in positive integers.

Question

Find all solutions to the diophantine equation $7^x=3^y+4$ in positive integers. I couldn't have much progress.

Clearly $(x,y)=(1,1)$ is a solution. And there's no solution for $y=2$.

Assume $y \ge 3$ and $x \ge 1$.

By $\mod 9$, we get $7^x \equiv 4\mod 9 \implies x \equiv 2 \mod 3 $.

By $\mod 7$,we get $y \equiv 1 \mod 6$.

I also tried $\mod 2$ but it didn't work.

Please post hints ( not a solution). Thanks in advance!

Answer 1

It's $3(3^a-1)=7(7^b-1)$ with $a=x-1$ and $b=y-1$.

Therefore $7\mid3^a-1$, so $a$ is a multiple of (what?).

Therefore, $3^a-1$ is a multiple of $13$.

Therefore, $7^b-1$ is a multiple of $13$.

Therefore, $b$ is a multiple of (what?).

Therefore, $7^b-1$ is a multiple of $9$.

Therefore, $3(3^a-1)$ is a multiple of $9$.

Therefore, $a$ is (what?).

Therefore, $x$ is (what?).

Answer 2

examples to study

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