Carmichael Numbers in Number Rings

for all integers a. This result gives us the rudimentary Fermat Compositeness Test: If a 6≡ a (mod n) for some integer a, then n is composite. While this has the advantage of being computationally simple, it has the distinct disadvantage of failing for some composite n and choice of a. Take, for example, n = 341 = 31 · 11 and a = 2. A quick computation tells us that 2 ≡ 2 (mod 341). Fortunately, we can also choose a = 3 to get 3 ≡ 168 (mod 341), thus proving that 341 is composite. We cannot always be so lucky. There are some composite n which fail this test no matter how we pick a. A Carmichael number is a composite integer n such that a ≡ a (mod n) for all integers a. The smallest such number is 561. The existence of Carmichael numbers means that the converse of Fermat’s Little Theorem fails. Even worse, the fact that there are infinitely many Carmichael numbers [1] means that the converse fails rather spectacularly. On the bright side, one can completely characterize all Carmichael numbers using Korselt’s Criterion.