What is the value of (v + f)? [Given: u is positive integer, 0 < f < 1, (√2+1)¹⁰ = u + f, (√2-1)¹⁰ = v]

Since (√2+1)¹⁰ + (√2-1)¹⁰ is an integer (call it N), we have u + f + v = N. Since 0 < f < 1 and 0 < v = (√2-1)¹⁰ < 1, we have 0 < f + v < 2. For u + (f+v) to be integer with u integer, f + v must be integer, hence f + v = 1.