On page 4, surely K and L are intended to be functions from the naturals to the naturals.
But, let z = 1001. Then the integral part of sqrt(z) = 31. Square that integral part and we get 961. Hence K(z) = 1001 - 961 = 40. Hence L(z) = 31 - 40 = -9. But L(z) should be positive so that we get the promised correspondence between N and N x N.