By Ciro Ciliberto, E. Laura Livorni, Andrew J. Sommese

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Otherwise output false. If Miller-Rabin outputs true for n, we can call it again with n and if it again outputs true then the probability that we have incorrectly determined that n is prime (when n is actually composite) decreases. Proof. We will prove that the algorithm is correct, but will prove nothing about how likely the algorithm is to assert that a composite is prime. We must prove that if the algorithm pronounces an integer n composite, then n really is composite. Thus suppose n is prime, yet the algorithm pronounces n composite.

8 (Reduction Map and Lift). We call the natural reduction map Z → Z/nZ, which sends a to a + nZ, reduction modulo n. We also say that a is a lift of a + nZ. , 7 is a lift of 1 mod 3, since 7 + 3Z = 1 + 3Z. 8). 9. A number n ∈ Z is divisible by 3 if and only if the sum of the digits of n is divisible by 3. Proof. Write n = a + 10b + 100c + · · · , where the digits of n are a, b, c, etc. Since 10 ≡ 1 (mod 3), n = a + 10b + 100c + · · · ≡ a + b + c + · · · (mod 3), from which the proposition follows.

Thus c = c as elements of (Z/mnZ)∗ . , that every element of (Z/mZ)∗ × (Z/nZ)∗ is of the form ψ(c) for some c. 2 implies that there exists c with c ≡ a (mod m) and c ≡ b (mod n). We may assume that 1 ≤ c ≤ nm, and since gcd(a, m) = 1 and gcd(b, n) = 1, we must have gcd(c, nm) = 1. Thus ψ(c) = (a, b). 6 (Multiplicative Function). A function f : N → C is multiplicative if, whenever m, n ∈ N and gcd(m, n) = 1, we have f (mn) = f (m) · f (n). 7 (Multiplicativity of ϕ). The function ϕ is multiplicative.

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