The negation would be "there exist a finite number of integers whose squareroot is irrational".

Just saying "there exist an infinite number of integers whose squareroot isrational" won't help because that istrue! There exist an infinite number of integers whose squareroot is rationalandan infinite number of integers whose squareroot is irrational.

But I don't know if a proof by contradiction is the best way to go. It would be sufficient to show that the squareroot of anyprimeis irrational.