Prime Numbers

3.4 Pseudoprimes 133
Theorem 3.4.4. For each integer a ≥ 2 there are infinitely many Fermat
pseudoprimes base a.
Proof. We shall show that if p is any odd prime not dividing a 2 − 1, then
n = a 2p − 1 / a 2 − 1 is a pseudoprime base a. For example, if a = 2 and
p = 5, then this formula gives n = 341. First note that
n = ap − 1
a − 1 · ap +1
a +1 ,
so that n is composite. Using (3.2) for the prime p we get upon squaring both
sides that a 2p ≡ a 2 (mod p). So p divides a 2p − a 2 .Sincep does not divide
a 2 − 1, by hypothesis, and since n − 1= a 2p − a 2 / a 2 − 1 , we conclude
that p divides n − 1. We can conclude a second fact about n − 1 as well: Using
the identity
n − 1 ≡ a 2p−2 + a 2p−4 + ···+ a 2 ,
we see that n − 1isthesumofanevennumberoftermsofthesameparity,
so n − 1 must be even. So far, we have learned that both 2 and p are divisors
of n − 1, so that 2p must likewise be a divisor. Then a 2p − 1 is a divisor of
a n−1 − 1. But a 2p − 1 is a multiple of n, so that (3.3) holds, as does (3.2). ✷
3.4.2 Carmichael numbers
In search of a simple and quick method of distinguishing prime numbers from
composite numbers, we might consider combining Fermat tests for various
bases a. For example, though 341 is a pseudoprime base 2, it is not a
pseudoprime base 3. And 91 is a base-3, but not a base-2 pseudoprime. Perhaps
there are no composites that are simultaneously pseudoprimes base 2 and 3,
or if such composites exist, perhaps there is some finite set of bases such that
there are no pseudoprimes to all the bases in the set. It would be nice if this
were true, since then it would be a simple computational matter to test for
primes.
However, the number 561 = 3 · 11 · 17 is not only a Fermat pseudoprime
to both bases 2 and 3, it is a pseudoprime to every base a. Itmaybeashock
that such numbers exist, but indeed they do. They were first discovered by
R. Carmichael in 1910, and it is after him that we name them.
Definition 3.4.5. A composite integer n for which a n ≡ a (mod n) for
every integer a is a Carmichael number.
It is easy to recognize a Carmichael number from its prime factorization.
Theorem 3.4.6 (Korselt criterion). An integer n is a Carmichael number
if and only if n is positive, composite, squarefree, and for each prime p dividing
n we have p − 1 dividing n − 1.
Remark. A. Korselt stated this criterion for Carmichael numbers in 1899,
eleven years before Carmichael came up with the first example. Perhaps