Strange equation

Algebra Level 3

x 12 + 1 = 4 x 4 x n − 1 \large x^{12}+1=4x^4\sqrt{x^n-1}

Find the smallest positive integer n n such that the above equation has real root.


The answer is 5.

Khang Nguyen Thanh by Khang Nguyen Thanh , 27, Vietnam

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1 solution

Chew-Seong Cheong
Apr 18, 2018

Let f ( x ) = x 12 + 1 − 4 x 4 x n − 1 f(x) = x^{12} + 1 - 4x^4\sqrt{x^n-1} . Let us find the minimum of f ( x ) f(x) .

d f ( x ) d x = 12 x 11 − 16 x 3 x n − 1 − 2 n x n + 3 x n − 1 = 12 x 11 x n − 1 − 16 x 3 ( x n − 1 ) − 2 n x n + 3 x n − 1 \begin{aligned} \frac {df(x)}{dx} & = 12x^{11} - 16x^3\sqrt{x^n-1} - \frac {2nx^{n+3}}{\sqrt{x^n-1}} \\ & = \frac {12x^{11}\sqrt{x^n-1}- 16x^3(x^n-1) - 2nx^{n+3}}{\sqrt{x^n-1}} \end{aligned}

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