A Bit of Number Theory
Fundamental Lemma of Number Theory: Let
be two integers. There exits unique integers
and
with
and
Note: Uniqueness follows from the fact that
implies
Definition: Let
be two integers. The greatest common divisor of
and
is defined to be largest integer
such that
divides both
and
.
Notation
.
Since
divides both
and
, we can use induction to show that
exists. For that matter, since
one can compute
(very inefficiently!) by
trying all values, starting at
and ending at the least of
and
Examples:
.
Lemma: Suppose
then
.
PROOF:
In fact all of their common divisors are in common!
The Euclidean Algorithm for computing
:
(See the proof below)
Theorem( Extended Euclidean Algorithm): Let
There exists integers
and
, computable using the Euclidean Algorithm such that
PROOF:
(By induction)
If
the theorem is trivial so for simplicity assume that
.
Again if
divides
the theorem is trivial so assume
hence
If
divides
then
and
.
Otherwise
hence
and
or
again if
divides
we are done else......
Example- Compute
Computing the simple-minded way requires
long divisions. Divide
and
by
through
.
Now by the euclidean algorithm,
This admittedly is an easy example but we see that EA requires
long divisions to find
.
Lemma: Suppose
is prime (the only divisors are
and
).
Suppose
divides
then
divides
or
divides
.
PROOF:
Suppose
does not divide
,
then
.
Thus there exists integers
and
, such that
Multiplying both sides of the equation by
gives
Since both terms on the left hand side are divisible by
so is the right hand side.
Simple Observation: Let
and
be primes. Let
.
The only divisors of
are
,
,
and
.
A Big Question: Suppose I know that
is the product of two primes
and
.
How hard is it to compute this primes? Again there is a simple-minded
answer.
.
Hence we only have to do at most
long divisions.
Suppose
has
digits, how long might that take?