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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.)

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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.

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