Let Q = n (n + 1). It is convenient to choose n = m2, for then Q is already a sum of two squares: Q = m2(m2 + 1) = (m2)2 + m2. If further m2 itself is a sum of two square, say m2 = p2 + q2, then
Q = (p2 + q2)(m2 + 1) = (pm + p)2 + (p - qm)2.
That the two representation or Q are distinct. Thus for example, we may take m = 5k, p = 3k, q = 4k, where k varies over natural numbers. In this case n = m2 = 25k2,
Q = (25k2)2 + (5K2)2 = (15k2 + 4k)2 + (20k2 - 3k)2.
Every k over natural number, we get infinitely many numbers the form n(n + 1) which can be expressed as a sum two squares in two distinct ways.