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Bounds on the number of elements representable as difference of squares/cubes

Source: Spain Mathematical Olympiad 2020 P6

July 15, 2020
difference of squaresnumber theorySpainAnalytic Number Theory

Problem Statement

Let SS be a finite set of integers. We define d2(S)d_2(S) and d3(S)d_3(S) as:
∙\bullet d2(S)d_2(S) is the number of elements a∈Sa \in S such that there exist x,y∈Zx, y \in \mathbb{Z} such that x2−y2=ax^2-y^2 = a ∙\bullet d3(S)d_3(S) is the number of elements a∈Sa \in S such that there exist x,y∈Zx, y \in \mathbb{Z} such that x3−y3=ax^3-y^3 = a
(a) Let mm be an integer and S={m,m+1,…,m+2019}S = \{m, m+1, \ldots, m+2019\}. Prove:
d2(S)>137d3(S)d_2(S) > \frac{13}{7} d_3(S)
(b) Let Sn={1,2,…,n}S_n = \{1, 2, \ldots, n\} with nn a positive integer. Prove that there exists a NN so that for all n>Nn > N:
d2(Sn)>4⋅d3(Sn) d_2(S_n) > 4 \cdot d_3(S_n)
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