I have difficulties with below problem:
Let and be primes with such that is not a factor of . As is a prime, we know that for any we can write in the form for some integer . Furthermore, since the greatest common divisor of and is , we can write for some integers and .
- Show that every integer can be written in the form for some integer and some integer .
- Prove the First Case of Fermat's Last Theorem for the exponents and . (That is, show that if there is a nonzero integer solution to for then is a factor of . and so on.)
Any help would be highly appreciated.