If P is a prime and x is a real number such that x2 P then

If P is a prime and x is a real number such that x^2 = P, then x is irrational.

Solution

Let, x be rational

So, x=r/s ,r,s integers, gcd(r,s)=1

SO,

(r/s)^2=P

r^2=Ps^2

Henec, P|r^2

P is prime so , P|r

Hence, r=kP for some integer k

k^2P^2=Ps^2

Pk^2=s^2

So, P|s

s=tP for some integer t

But, gcd(r,s)=1

HEnce a contradiction

HEnce, x is irrational