grandes-ecoles 2014 QII.B.1

grandes-ecoles · France · centrale-maths2__mp Number Theory GCD, LCM, and Coprimality

Throughout this subsection II.B, we fix two natural integers $m$ and $n$. The Chebyshev polynomials of the second kind $U_n$ have roots $\cos\left(\frac{k\pi}{n+2}\right)$ for $k = 1, \ldots, n+1$.
Let $h$ be the $\gcd$ in $\mathbb{N}$ of $m+1$ and $n+1$. By examining the common roots of $U_n$ and $U_m$, show that $U_{h-1}$ is a $\gcd$ in $\mathbb{R}[X]$ of $U_n$ and $U_m$.

Throughout this subsection II.B, we fix two natural integers $m$ and $n$. The Chebyshev polynomials of the second kind $U_n$ have roots $\cos\left(\frac{k\pi}{n+2}\right)$ for $k = 1, \ldots, n+1$.

Let $h$ be the $\gcd$ in $\mathbb{N}$ of $m+1$ and $n+1$. By examining the common roots of $U_n$ and $U_m$, show that $U_{h-1}$ is a $\gcd$ in $\mathbb{R}[X]$ of $U_n$ and $U_m$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4