Prime Numbers

3.8 Exercises 165
3.12. Show that a composite number n is a Carmichael number if and only
if a n−1 ≡ 1(modn) for all integers a coprime to n.
3.13. [Beeger] Show that if p is a prime, then there are at most finitely many
Carmichael numbers with second largest prime factor p.
3.14. For any positive integer n let
F(n) = a (mod n) :a n−1 ≡ 1(modn) .
(1) Show that F(n) is a subgroup of Z ∗ n, the full group of reduced residues
modulo n, and that it is a proper subgroup if and only if n is a composite
that is not a Carmichael number.
(2) [Monier, Baillie–Wagstaff] Let F (n) =#F(n). Show that
F (n) =
gcd(p − 1,n− 1).
p|n
(3) Let F0(n) denote the number of residues a (mod n) such that an ≡ a
(mod n). Find a formula, as in (2) above, for F0(n). Show that if
F0(n)