Wanna Practice newton's Sum
If are roots of the equation above, find the last 6 digits of the expression below.
The answer is 532127.
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1 solution
By Vieta's formulas, we have p + q = 1 3 and ( p q = 7 ) . Then we have:
\(\begin{array} {} p^2+q^2 & = (p+q)^2 - 2pq & = 13^2 - 2(7) & = 155 \\ p^4+q^4 & = (p^2+q^2)^2 - 2(pq)^2 & = 155^2 - 2(49) & = 24123 \\ p^8+q^8 & = (p^4+q^4)^2 - 2(pq)^4 & = 24123^2 - 2(2401) & = 572496527 \\ p^{16}+q^{16} & = (p^8+q^8)^2 - 2(pq)^8 & = 572496527^2 - 2(5764801) & = 327752273415 \boxed{532127} \end{array} \)