A Bit More Number Theory
Notation: Let
be three integers . We write
if
divides
.
Note that from the previous section we have
implies
or
Lemma:
Let
,
prime, then there exists an
integer
such that
PROOF:
A simple counting argument shows that there are integers
with
.
Assume
.
We have
.
But since
we have
.
Lemma: Let
then
.
PROOF: Let
be the least integer such that
.
It suffices to show that
divides
.
To do this consider the set of integers.
,
We can divide this set into disjoint subsets of
elements using the relationship
iff
for some
.
Observations: The subsets are of the form
.
The other points is that
depends on the
.
For example, let
and
.
Note
.
On the other hand, if
and
, we have
Of course,
.
Theorem: Let
and
be primes. Let
.
Let
be relatively prime to both
and
, hence
,
then
.
Hence,
.
PROOF: Both
and
, divide
.
(
)
Observation: Another way of looking at this is
that
and
Of course, the Extended Euclidean Algorithm tells us that
can
be computed for any relatively prime pair