Question

Prove that there are infinitely many positive integers n such that n (n + 1) can be expressed as a sum of two positive squares in at least two different ways. (Here a² + b² and b² + a² are considered as the same representation.)

Answer 1

Explanation:

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.