Super fun complex number question

Algebra Level 3

Suppose z z is a complex number such that z 5 + 1 = 0. z^5 + 1 = 0.

If z 4 + z 2 + 1 = z m + z n z^4 + z^2 + 1 = z^m + z^n , then find ∣ m − n ∣ \left|m - n\right| , where m m and n n are the smallest possible positive integers for which the listed equation is true.


The answer is 2.

Harry Obey by harry obey , 21, Australia

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1 solution

Tom Engelsman
Jul 21, 2019

Let z 5 + 1 = ( z + 1 ) ( z 4 − z 3 + z 2 − z + 1 ) = 0 z^5 + 1 = (z+1)(z^4 - z^3 + z^2 - z + 1) = 0 . If the quantity z 4 + z 2 + 1 = z m + z n z^4 + z^2 + 1 = z^{m} + z^{n} holds true, and we desire m , n m, n to be the smallest possible positive integers, then we can now write:

( z + 1 ) ( z 4 − z 3 + z 2 − z + 1 ) = ( z + 1 ) ( z m + z n − z 3 − z ) = 0 ⇒ ( m , n ) = ( 1 , 3 ) ; ( 3 , 1 ) (z+1)(z^4 - z^3 + z^2 - z + 1) = (z+1)(z^{m} + z^{n} - z^3 - z) = 0 \Rightarrow (m ,n) = (1,3); (3,1) .

Hence, ∣ m − n ∣ = ∣ 3 − 1 ∣ = ∣ 1 − 3 ∣ = 2 . |m - n| = |3 - 1| = |1 - 3| = \boxed{2}.

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