Fermat’s Test

Fermat’s little theorem says for prime p that ap−1 ≡ 1 mod p for all a 6≡ 0 mod p. A naive extension of this to a composite modulus n ≥ 2 would be: for a 6≡ 0 mod n, an−1 ≡ 1 mod n. Let’s call this “Fermat’s little congruence.” It may or may not be true. When n is prime, it is true for all a 6≡ 0 mod n. But when n is composite it usually has many counterexamples. Example 1.1. When n = 15, the table below shows that for only four a 6≡ 0 mod 15 do we have a14 ≡ 1 mod 15. a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a14 mod 15 1 4 9 1 10 6 4 4 6 10 1 9 4 1 Example 1.2. Among all the numbers a 6≡ 0 mod 91, 36 of them (less than half) satisfy a90 ≡ 1 mod 91. The situation for small a is shown below. a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 · · · a90 mod 91 1 64 1 1 64 64 77 64 1 1 64 1 78 14 64 · · · If Fermat’s little congruence an−1 ≡ 1 mod n fails for even one a 6≡ 0 mod n, then n isn’t prime, so it’s composite. For instance, from above 214 6≡ 1 mod 15 and 290 6≡ 1 mod 91, so 15 and 91 are composite. Of course 15 and 91 are small enough that their compositeness can be seen by direct factoring (15 = 3 ·5 and 91 = 7 ·13). The real significance of breaking Fermat’s little congruence is for much larger n, since it lets us prove a large number is composite without having to factor it. This is what we will explore here.