Orders Modulo A Prime

In this article I develop the notion of the order of an element modulo n, and use it to prove the famous n 2 + 1 lemma as well as a generalization to arbitrary cyclotomic polynomials. References used in preparing this article are included in the last page. §1 Introduction I might as well state one of the main results of this article up front, so the following discussion seems a little more motivated. Theorem 1.1 Let p be an odd prime. Then there exists an integer n such that p | n 2 + 1 if and only if p ≡ 1 (mod 4). By introducing the notion of order, we will prove that p | n 2 + 1 =⇒ p ≡ 1 (mod 4). By introducing the notion of a primitive root, we will prove the converse direction. Finally, we will write down the generalized version of this n 2 + 1 lemma using cyclotomic polynomials. §2 Orders Let p be a prime and take a ≡ 0 (mod p). The order 1 of a (mod p) is defined to be the smallest positive integer m such that a m ≡ 1 (mod p). This order is clearly finite because Fermat's Little Theorem tells us a p−1 ≡ 1 (mod p), id est, the order of a is at most p − 1. Exhibited below are the orders of each a (mod 11) and a (mod 13).