Prove that there is no such that .
Proof. Suppose otherwise, that there is some rational number such that . Then, since , we know there exist integers not both even (I.3.12, Exercise #10 (e) such that . Then, since we have
But, by I.3.12, Exercise #10 (d), we know implies both and are even, contradicting our choice of and not both even. Hence, there can be no such rational number