Euler's phi function is another number theoretic function, and an extremely important one as it allows us to generalize Fermat's Little Theorem (details in section 7.3). Leonhard Euler (1707-1783) is really quite an amazing character, and hopefully you enjoyed the description given of his life in section 7.1. Please do read the historical notes, and remember some of these stories!
Definition 7.1: For 
, let 
denote the number of
positive integers not exceeding n that are relatively prime to n.
Problem: compute
-
-
-
-
, p prime
Theorem 7.1: If p is prime and k>0, then
Lemma: Given integers a, b, c,
if and only if
and
.
Theorem 7.2: The function 
is a multiplicative function.
Theorem 7.3: If the integer n>1 has the prime factorization
, then
Proof: is a multiplicative function!
Theorem 7.4: For n>2, is an even integer.