MUHAMMAD YUSRAN BASRI
1211441023
MATH ICP B 2012
FAKULTAS MIPA UNIVERSITAS NEGERI MAKASSAR
SUMMARY
Fermat’s Little Theorem and Euler’s Theorem
From the last few results, we note that for a given positive integer m, it is useful to
consider the congruence classes that are relatively prime to m. For any positive integer m we
denote by ϕ(m) the number of all positive integers n less than m that are relatively prime to m.
The function ϕ is called Euler’s totient function. It is clear that ϕ(1) = 1 and for any prime p,
ϕ(p) = p − 1. Moreover, if n is a positive integer such that ϕ(n) = n − 1, then n is a prime.
A set S of integers is also called a reduced complete set of residue classes modulo m if for each
i with 0 ≤ i ≤ n −1 and gcd(i,m) = 1, there is an element s ∈ S such that i ≡ s (mod m). It is clear
that a reduced complete set of residue classes modulo m consists of ϕ(m) elements.
Theorem 1.28. [Euler’s Theorem] Let a and m be relatively prime positive integers. Then
aϕ(m) ≡ 1 (mod m).
Proof: Consider the set S = {a1, a2, . . . , aϕ(m)} consisting of all positive integers less than m that
are relatively prime to m. Because gcd(a, n) = 1, it follows from Proposition 1.26 that
{aa1, aa2, . . . , aaϕ(m)}
is another reduced complete set of residue classes modulo n. Then
(aa1)(aa2) · · · (aaϕ(n)) ≡ a1a2 · · · aϕ(n) (mod m).
Using that gcd(ak , n) = 1, k = 1, 2, . . . , ϕ(n), the conclusion now follows. Setting m = p as
prime, Euler’s theorem becomes Fermat’s little theorem.
Theorem 1.29. [Fermat’s Little Theorem] Let a be a positive integer and let p be a prime. Then
a p ≡ a (mod p).
Proof: We present an alternative proof independent of Euler’s theorem. We induct on a. For a =
1 everything is clear. Assume that p | (a p − a). Then
𝑝
Using the fact that p | ( ) for 1 ≤ k ≤ p − 1 (Corollary 1.10) and the inductive hypothesis, it
𝑘
follows that p divides (a+1)p−(a+1); that is, (a+1)p ≡ (a+1) (mod p).
Clearly, Fermat’s little theorem is a special case of Euler’s theorem. But with a few more
properties on the Euler function ϕ that we will develop, we can derive Euler’s theorem from
Fermat’s little theorem Note also another form of Fermat’s little theorem:
Let a be a positive integer relatively prime to prime p. Then a p−1 ≡ 1 (mod p). Next, we present a
few examples involving these two important theorems.