By D. Mumford

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Is a perfect square. 14 Prove that a positive integer n√is prime if and only if n is not divisible by any prime p with 1 < p ≤ n. 2 The Ring of Integers Modulo n A startling fact about numbers is that it takes less than a second to decide with near certainty whether or not any given 1,000 digit number n is a prime, without actually factoring n. The algorithm for this involves doing some arithmetic with n that works differently depending on whether n is prime or composite. In particular, we do arithmetic with the set (in fact, “ring”) of integers {0, 1, .

The Mod(x,n) command returns the equivalence class of x in Z/nZ. 3 Wilson’s Theorem The following characterization of prime numbers, from the 1770s, is called “Wilson’s Theorem,” though it was first proved by Lagrange. 22 (Wilson’s Theorem). An integer p > 1 is prime if and only if (p − 1)! ≡ −1 (mod p). For example, if p = 3, then (p − 1)! = 2 ≡ −1 (mod 3). If p = 17, then (p − 1)! = 20922789888000 ≡ −1 (mod 17). But if p = 15, then (p − 1)! = 87178291200 ≡ 0 (mod 15), so 15 is composite. Thus Wilson’s theorem could be viewed as a primality test, though, from a computational point of view, it is probably one of the world’s least efficient primality tests since computing (n−1)!

7 repeatedly, we see that a = a1 a2 · · · ar has order q1n1 · · · qrnr = p − 1, so a is a primitive root modulo p. 9. 8 when p = 13. We have p − 1 = 12 = 22 · 3. The polynomial x4 − 1 has roots {1, 5, 8, 12} and x2 − 1 has roots {1, 12}, so we may take a1 = 5. The polynomial x3 − 1 has roots {1, 3, 9}, and we set a2 = 3. Then a = 5 · 3 = 15 ≡ 2 is a primitive root. To verify this, note that the successive powers of 2 (mod 13) are 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1. 10. 8 is false if, for example, p is replaced by a power of 2 bigger than 4.

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