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A nice limit

Source: Brazil Undergrad MO 2021

March 15, 2021
limitscalculusBrazilian Undergrad MOBrazilian Undergrad MO 2020geometry

Problem Statement

Let R>0R > 0, be an integer, and let n(R)n(R) be the number um triples (x,y,z)∈Z3(x, y, z) \in \mathbb{Z}^3 such that 2x2+3y2+5z2=R2x^2+3y^2+5z^2 = R. What is the value of
lim⁡R→∞n(1)+n(2)+⋯+n(R)R3/2\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}?
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