There isn’t too much going in this question.Nonetheless,I just liked it for some bloody reason.
- Let and be positive integers where .If is prime,prove that is even and n is of the form where
where is an odd positive integer.
Simply use the factorization formula to prove this.
Since ,it is quite clear that ,this means that is odd and so is even
Now,to prove the second claim,assume that is not a power of 2.
This means that some odd prime such that ie where k is a positive integer.
Here is an odd positive integer so by the Lemma,
This even accounts for the case when is prime ie
Here the factor ,so it contradicts the fact that is prime.
This completes the proof.