User:Foxjwill/Proofs
Proof that
, where
is prime, is irrational
- Assume that
is rational. Then
such that
is coprime to
and
.
- It follows that
, and that
.
- So, by the properties of exponents along with the unique factorization theorem,
divides both
and
.
- Factoring out
from (2), we have
for some
.
- Therefore
divides
.
- But this contradicts the assumption that
and
are coprime.
- Therefore
.
- Q.E.D.
A theorem
THEOREM. Let be a circle of radius
, let
be the set of chords of
, for all
, let
. Then for all
, there exists an angle
such that for all
, there exists a positive integer
such that for all sets