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