suppose x^2+x+1 divides x^a+x^b+1
let the roots of x^2+x+1 be r1, r2
then r1^a+r1^b+1=r2^a+r2^b+1=0 by the factor theorem
iff r1^a+r2^a+r1^b+r2^b+2=r1^a-r2^a+r1^b-r2^b=0

let F(x)=r1^x+r2^x, G(x)=r1^x-r2^x
then F(x+2)+F(x+1)+F(x)=0, G(x+2)+G(x+1)+G(x)=0 (characteristic equation)

also F(0)=2, G(0)=0, F(1)=-1, G(1)=3i
so F would be 2,-1,-1 repeating, F would be 0,3i,-3i repeating
notice we have F(a)+F(b)+2=G(a)+G(b)=0
so its obvious that a=1 (mod 3), b=2 (mod 3)
or a=2 (mod 3), b=1 (mod 3)

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