The most important result in this section is ``Fermat's Little Theorem''. This is one of the crucial results which has made number theory so valuable in recent years (in cryptography). Number theory was beloved of Hardy because he thought it practically useless - how wrong could he be!
Theorem 5.1 (Fermat's Theorem): Let p be prime and suppose that p
does not divide a. Then 
.
Corollary: If p is a prime, then 
for any integer
a.
Example#2c, p. 94 :
The corollary is a generalization of Fermat's little theorem, which obviates the need to include the divisibility criterion. At times, though, it's really Fermat's little theorem that one wants to use, since it's nice to get large powers to work out to 1....
The proof of the corollary by induction is really interesting! It's surprising that induction would work here, perhaps - at least, it surprised me.
Lemma: If p and q are distinct primes with 
and
, then 
.
The rest of the section deals with numbers that have primal pretentions: pseudoprimes, and pseudoprimes to a base a, and absolute pseudoprimes.
pseudoprime: a composite number n such that 
.
A Chinese theorem of 2500 years ago speculated that numbers that so divide are
prime, and that primes so divide. It was proven wrong by counterexample (341),
in 1819 ( 
).
pseudoprime to the base a: more generally, a composite number n such
that 
.
absolute pseudoprime: a composite number n which satisfies
for all integers a.
Theorem 5.2: If n is an odd pseudoprime, then 
is a larger
one.
Theorem 5.3: Let n be a composite square-free integer, say,
, where the 
are distinct primes. If 
for
, then n is an absolute pseudoprime.
So absolute pseudoprimes behave like primes, as far as Fermat's Little theorem is concerned. Fermat's little theorem couldn't detect them as pretenders.