Problem

Source: Spain Mathematical Olympiad 2020 P6

Tags: difference of squares, number theory, Spain, Analytic Number Theory



Let S be a finite set of integers. We define d2(S) and d3(S) as: d2(S) is the number of elements aS such that there exist x,yZ such that x2y2=a d3(S) is the number of elements aS such that there exist x,yZ such that x3y3=a (a) Let m be an integer and S={m,m+1,,m+2019}. Prove: d2(S)>137d3(S) (b) Let Sn={1,2,,n} with n a positive integer. Prove that there exists a N so that for all n>N: d2(Sn)>4d3(Sn)