You cannot use any theorems involving modular arithmetic.
1.) Prove the theorem that "Let such that or , we get that "
Note: is the greatest common divisor.
2.) Let be a prime number. Prove that if and only if .
3.) Prove that if and are prime numbers, then is also prime number.
4.) Prove that there exists infinitely many positive integer such that is composite.
5.) Find all such primes and positive numbers such that are perfect squares.
This is the part of Thailand 1st round math POSN problems.