Positive Integers to make a square How many integers $n$  make the expression $7^n + 7^3 + 2\cdot7^2$ a perfect square?
Factoring $7^2$ we have that $7^n + 7^3 + 2\cdot7^2 = 7^2\cdot(7^{n-2} + 9)$.
How do we prove that the 2nd factor only has $1$ solution when $n = 3$?

An "equivalent" way of doing things, is to note that $7^3 + 2 \times 7^2 = 343+98=441 = 21^2$.
Now, if $7^n + 21^2 = y^2$ for some $y$, then $7^n = (y-21)(y+21)$.
So the prime factorization of $(y-21)$ and $(y+21)$ can both contain only the prime $7$, by unique factorization theorem. Therefore, $y - 21$ and $y+21$ are both powers of $7$. The difference between these powers of $7$ is $y+21 - (y-21) = 42$.
Only two such powers have a difference of $42$, which are $49$ and $7$.This is not very difficult to see from the fact that $7^n - 7^m = 7^{n-m}(7^{m} - 1) = 42$ forces $n-m = 1$ and $m = 1$. Therefore, $(y-21)(y+21) = 7 \times 49 = 343$, so $n = 3$ is forced from here.

Thinking of it, you could also have done $7^{n-2} + 9 = x^2$ since you factorized out $7^2$, but the same logic will apply, since $9 = 3^2$.
