A prime number (excluding 2 and 3) must have form 6k+1 or 6k+5. Let p have form 3k+1. If p had form 6k+5 then 6k+5=3(2k+1)+2 so p would have form 3k+2, a contradiction. Thus, p must have form 3k+1.

Let x be a positive integer of the form 3k+2. Since it is odd we know x is a product of odd prime numbers. Thus, each prime factor has form 3k+1 or 3k+2. If each prime factor had form 3k+1 then the product of all of them still will have form 3k+1, which is a contradiction.2. Show that any integer of the form 3n+2 must have a prime factor of the same form.

Hint:3. Show that 7 is the only prime of the form n^3-1

We will asume that . If by unique factorization it forces . Thus, is the solution.4. Show that p=5 is the only prime for which 3p+1 is a perfect square.

If then then .5. If p is prime and p divides a^n, show that p^n divides a^n

See this.6. Show that every integer of the form n^4+4 is composite