$-$ Trigonometry $+$ and $-$ Polynomials?

Algebra Level 5

This question is a follow up from Joel's problem regarding something very interesting I found.

For every positive integer $n$ , $\cos{(nx)}$ can be written as a polynomial in $\cos{(x)}$ with integral coefficients. We would call it ${ P }_{ n }(y)$ , where $y=\cos{(x)}$ . For example, we have ${ P }_{ 1 }(y)=y, { P }_{ 2 }(y)=2{ y }^{ 2 }-1$

Denote $s(n)$ to be the sum of values of the coefficients of ${ P }_{ n }(y)$ . For example, $s(1)=1,\quad s(2)=1$

Find the smallest positive integer value of $A$ such that for all integers $n>0$ , $s(n)$ and $s(n+A)$ are congruent modulo $1234$ . Input $A \times 3$ as your answer.

To avoid ambiguity, assume $k>0$ if $k$ is a positive integer.

The answer is 3.

2 solutions

Pi Han Goh
Mar 12, 2015

Essentially this just boils down to stating $\cos (nx)$ in terms of a polynomial of $\cos (x)$

Let $x = 0$ , $1 = \cos(0) = \text{Sum of coefficients} \Rightarrow A = 1$

Parth Kohli
Jan 24, 2015

Pardon me, but the only plausible way I found was pattern recognition using a few identities I had committed to memory. I'm still looking for a reason as to why this happens.

$\cos(x) = \cos(x) \Rightarrow s(1) = 1$ .

$\cos(2x) = 2\cos^2 x - 1 \Rightarrow s(2) = 1$ .

$\cos(3x) = 4\cos^3 x - 3 \cos (x) \Rightarrow s(3) = 1$ .

$\cos(4x) = 8 \cos^4 x - 8 \cos^4 x + 1 \Rightarrow s(4) = 1$ .

I jumped to the conclusion that $s(n) = 1$ for all integral values of $n$ . So we can simply say that $A$ can take any integral value, but since we restrict it to positive integers, $A_{\rm min} = 1 \Rightarrow 3A = \boxed{3}$ .

Update: This document also confirms my suspicion.

Parth Kohli - 6 years, 4 months ago

− - − Trigonometry + + + and − - − Polynomials?

The answer is 3.

2 solutions

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$-$ Trigonometry $+$ and $-$ Polynomials?