Problem

Source: 2018 Brazil 4th TST Day 2 #5

Tags: algebra, polynomial



Prove: there are polynomials S1,S2, in the variables x1,x2,,y1,y2, with integer coefficients satisfying, for every integer n1, dndSn/dd=dnd(xn/dd+yn/dd)()Here, the sums run through the positive divisors d of n. For example, the first two polynomials are S1=x1+y1 and S2=x2+y2x1y1, which verify identity () for n=2: S21+2S2=(x21+y21)+2(x2+y2).