Chebyshev polynomial and prime number

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0 replies

jlacosta

Apr 2, 2025

0 replies

Why is this series not the Fourier series of some Riemann integrable function

tohill 1

N a minute ago by alexheinis

$\sum_{n=1}^{\infty}{\frac{\sin nx}{\sqrt{n}}}$ (0<x<2π)

1 reply

tohill

Today at 8:08 AM

alexheinis

a minute ago

Research Opportunity

dinowc 0

2 hours ago

Hi everyone, my name is William Chang and I'm a second year phd student at UCLA studying applied math. Over the past year, I've mentored many undergraduates at UCLA to finished papers (currently under review) in reinforcement learning (see here. :juggle:

I'm looking to expand my group (and the topics I'm studying) so if you're interested, please let me know. I would especially encourage you to reach out to me chang314@g.ucla.edu if you like math. :wow:

0 replies

dinowc

2 hours ago

0 replies

Computational Calculus - SMT 2025

Munmun5 3

N 2 hours ago by alexheinis

Source: SMT 2025

1. Consider the set of all continuous and infinitely differentiable functions $f$ with domain $[0,2025]$ satisfying $f(0)=0,f'(0)=0,f'(2025)=1$ and $f''$ is strictly increasing on $[0,2025]$ Compute smallest real M such that all functions in this set , $f(2025)<M$ .
2. Polynomials $A(x)=ax^3+abx^2-4x-c$ $B(x)=bx^3+bcx^2-6x-a$ $C(x)=cx^3+cax^2-9x-b$ have local extrema at $b,c,a$ respectively. find $abc$ . Here $a,b,c$ are constants .
3. Let $R$ be the region in the complex plane enclosed by curve $f(x)=e^{ix}+e^{2ix}+\frac{e^{3ix}}{3}$ for $0\leq x\leq 2\pi$ . Compute perimeter of $R$ .

3 replies

Munmun5

Today at 9:35 AM

alexheinis

2 hours ago

x^{2s}+x^{2s-1}+...+x+1 irreducible over $F_2$?

khanh20 1

N 6 hours ago by khanh20

With $s\in \mathbb{Z}^+; s\ge 2$ , whether or not the polynomial $P(x)=x^{2s}+x^{2s-1}+...+x+1$ irreducible over $F_2$ ?

1 reply

khanh20

6 hours ago

khanh20

6 hours ago

No more topics!

Chebyshev polynomial and prime number

mofidy 2

N Apr 10, 2025 by mofidy

Let $U_n(x)$ be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could $U_n(x) -1$ be a prime number?
Thanks.

2 replies

mofidy

Apr 3, 2025

mofidy

Apr 10, 2025

Chebyshev polynomial and prime number

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mofidy

132 posts

mofidy

#1 h Apr 3, 2025, 5:51 PM

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Let $U_n(x)$ be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could $U_n(x) -1$ be a prime number?
Thanks.

This post has been edited 1 time. Last edited by mofidy, Apr 3, 2025, 8:15 PM

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Snoop76

322 posts

Snoop76

#2 h Apr 4, 2025, 6:33 AM

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You can prove that:

For $n$ even $U_n(x)-1$ is even .

For $n=2p-1$ odd we can check that $U_{2p-1}(x)-1=0$ for $x=cos\frac{2k\pi}{2p-1}$ and $U_{2p-1}(x)-1=0$ for $x=cos\frac{(2k+1)\pi}{2p+1}$

From here results that $U_{2p-1}(x)-1$ is divisible with the minimal polynomial of $cos\frac{2k\pi}{2p-1}$ and $cos\frac{(2k+1)\pi}{2p+1}$ .

So $U_{2p-1}(x)-1$ can be factorised to at least two irreductibil polynomials.

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mofidy

132 posts

mofidy

#3 h Apr 10, 2025, 2:43 PM

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Thanks @Snoop76

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