What numbers have this property?
Given that x 2 0 1 5 + y 2 0 1 5 = − 1 and x 2 0 1 6 + y 2 0 1 6 = 2 , find x + y .
The answer is -1.
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1 solution
You still need to prove that ω , ω 2 are the only solutions for x and y .
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Let x and y be the roots of a polynomial P ( x ) , then, obviously, P ( x ) is of degree 2. By Newton's sums, eventually we can find the coefficients of P ( x ) , and since we have two sums of degrees 2 0 1 5 and 2 0 1 6 respectively, there is one and only one polynomial (with leading coefficient 1) with roots x and y , and that would be P ( x ) = x 2 − x + 1 , so there is only one set of solutions for x and y ...
Will that suffice? Or is something wrong with the solution...?
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No you can't make that conclusion. You're only given two equations, but that does not necessarily means that there is only one solution of x+y.
Counterexample: x^3+y^3=-1, x^4+y^4=2, but x+y have many solutions.
One can observe that cube roots of unity have the property ω n + ( ω 2 ) n = − 1 as long as n is not divisible by 3 , and ω n + ( ω 2 ) n = 2 as long as n is divisible by 3 . Therefore, x and y are cube roots of unity, and since x + y = x 1 + y 1 , and 1 is not divisible by 3, our answer is − 1