The set of coprimes of n, where n is an integer larger than 1, is an
infinite set, but considered modulo n, it's a set whose size is the totient of
n, φ(n). In this section, we will consider everything using
modulo arithmetic. I'll try to put a little notice to that effect at the
top of every page.
First, let's start with an example. The set of coprimes of 12 is
{1,5,7,11}. Remember, though numbers like 13 and 145 are coprime to 12,
these are equivalent to 1 (mod 12), so we don't list them separately.
The product of any two coprimes of n is another coprime of n
Proof: Suppose the opposite -- that ab shares a factor, f, with n, and
f>1, even though GCD(a,n)=1 and GCD(b,n)=1. Since f is larger than
one, it has a prime factor, p. So ab shares this prime factor, p, with
n. If p|ab then p|a or p|b, so either GCD(a,n) is not 1 or GCD(b,n) is
not 1, a contradiction.
If a, b, c are coprimes of n, and b is not equal to c, then ab is not
equal to ac
Proof: Suppose a is coprime to n, and b and c are two different numbers,
also coprime to n, in the range [1,n-1] and ab=ac, mod n.
Then ab-ac=nk, where k is an integer.
a(b-c)=nk, and since b-c is smaller than n in absolute value, a must be
larger than k in absolute value.
a and n have no factors in common, so a|k, but a can't divide k, since a is
larger than k, a contradiction.
. . . . . .
Squares.
Fermat's Theorems -- Fermat's
Little Theorem, in particular.
Group - definition of a "group",
which has properties closure, associativity, identity, and inverse.
The proof that primes of the form
4k+3 can be expressed as the sum of four squares
uses Fermat's Little Theorem.
A bunch of Ring Theory theorems
and definitions.
Proof that
there is no solution to the Diophantine equation A^{4}+B^{4}=C^{2},
which means Fermat's last theorem is true