We must have since 1 is a perfect square. Thusncan be written as a product of distinct primes: .

Now if all the ’s were even,nwould be a perfect square. Since it isn’t, at least one of the ’s must be odd. Hence, we can write wheremis odd andpdoes not dividek.

Suppose is rational: , where , and .

Then

aspis prime as .

Hence the factor ofpoccurs an odd number of times in . This is a contradiction because is a perfect square and so each prime factor of has to occur an even number of times.

Thus must be irrational.