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On estimation of the number of solutions

Source: 2021 China TST, Test 2, Day 1 P3

March 21, 2021
algebranumber theoryDiophantine equation

Problem Statement

Given positive integers a,b,ca,b,c which are pairwise coprime. Let f(n)f(n) denotes the number of the non-negative integer solution (x,y,z)(x,y,z) to the equation ax+by+cz=n.ax+by+cz=n. Prove that there exists constants α,β,γ∈R\alpha, \beta, \gamma \in \mathbb{R} such that for any non-negative integer nn, ∣f(n)−(αn2+βn+γ)∣<112(a+b+c).|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).
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