Notes on Explorations
E.l. The identity
u(ud + bc) + (2bd + c”) = c(c + ub) + d(2b + u”)is useful in picking the coefficients of t4 + at3 + bt2 + ct +d in such a way that
four coefficients of the nine in the square of this polynomial will vanish. The
polynomial
t4 + 2t3 - 2t2 + 4t + 4
has five terms in its square.
E.2. Write {ui} = (m) {bi} if Cuf = Cbf (k = 0, 1, 2, 3, 4,... , m). It can
be seen that, if {ui} = (m) {bi}, then {UU~ + b} = (m) {ubi + V} for any u
and v. To show how to construct pairs of subsets of integers with the first
few powers equal, we illustrate with an example how to move from simple
sets to more complex sets:
(1) j-1, 11 = (1) i-2,21(2) {1,3) = (1) {0,4)(3) {4,0,4) = (2) {-2,2,1,3)(4) {--LO, 4) = (2) i--2,2,3)(5) i-2, -1,3) = (2) {--3,1,21(6) {3,4,8) = (2) {2,6,7)
(7) {-2,-l&3,2,6,7)=(3)(-3,1,2,3,4,8}(8) j-2, -1,677) = (3) {-3,1,4,8)(9) (-9, -7,7,9) = (3) {-ll,-3,3,11).For the problem posed in (d), a solution for d = 3, m = 2 is{1,6,8,12,14,16,20,22,27)= (2) {2,4,9,10,15,17,21,23,25)
= (2) {3,5,7,11,13,18,19,24,26).