Thread: Question #1: Prime

If p is an odd prime, show that p is of the form.

(a) 4n + 1 or 4n+3 for some n.

(b) 6n + 1 or 6n + 5 for some n.

(a) is straightforward. If p is odd, then it cannot be a multiple of 2, which rules out 4n and 4n + 2. Then p must be either 4n + 1 or 4n + 3.

(b) is a little harder but not much. If p is odd, then it is either 6n + 1, 6n + 3, or 6n + 5. However, 6n + 3 = 3(2n + 1) and is not prime. Therefore, p is either 6n + 1 or 6n + 5.