The minimum k is the denominator of the supposed irreducible fraction p/q=sqrt(2). He is stating that then there is a different irreducible fraction r/s=sqrt(2). This is because you would have

p^2=2q^2

with p and q mutually prime, which is impossible.

Instead of irreducible fractions with coprime numbers, he is doing the “dual” using the minimum possible value of the denominator.