Good day to the MHF. Here's my problem

Find all integer solutions to

Just making sure my solution is complete

Solution

By CRT, we can break this up to

And now we deal with each separately

For

we use FLT to simplify:

Simplifying

. Solves to give

Now dealing with

FLT:

Simplifying

, we solve this via guesswork

Let

Solution:

So now we use CRT to solve

Solve to get

So

And a solution to our original equation is

Is this all the solutions or am I missing some? If so, where is te error in my method preventing me from obtaining the others?

Thanks in advance for the help.