let [root(n²+1)]=[root(n²+λ)] , where n, λ€N . show that λ can have 2n different values.

for 1≤ λ≤ 2n we have n²< n²+λ<(n+1)²

Hence n < √(n²+λ) < (n+1) and hence we have [√(n²+λ)] = [√(n²+1)] = n

If λ≤0 or λ>2n,  [√(n²+λ)] ≠ n

So these are the 2n values of λ for which the given equation holds