1
vote
0answers
13 views

Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
0
votes
0answers
10 views

Monic polynomials in $(\mathbb Z/p\mathbb Z)[X]$ [duplicate]

Be $p$ a prime number. how many monic polynomials degree 2 there are in $(\mathbb Z/p\mathbb Z)[X]$. How many of them are reducible and how many are not? What I tried to do is, let $f(x)$ be ...
1
vote
2answers
50 views

$\mathbb{Q}$ adjoining primes and the sum of root of those primes

I have $p$, $q$ as primes, and I want to show that $\mathbb{Q}(\sqrt{p},\sqrt{q})=\mathbb{Q}(\sqrt{p}+\sqrt{q})$. I was thinking about using inclusion both ways, so what does an element in ...
1
vote
2answers
69 views

Finding a least common multiple (LCM)

My Algebra 2 book explains how to find a least common multiple: Find the least common multiple of $4x^2 - 16$ and $6x^2 - 24x + 24$. Solution Step 1 Factor each polynomial. Write ...
5
votes
2answers
99 views

Polynomial is prime when evaluated at prime numbers

Let $P(x)$ be a polynomial with integer coefficients such that $P(p)$ is prime for all prime $p$. What are all possible polynomials $P(x)$? Certainly $P(x)=x$ and $P(x)=p$ with $p$ prime satisfy that ...
4
votes
2answers
93 views

Finding a function whose value at $n$ is the $n^{\text{th}}$ prime

For positive integers $a$ and $b$, evaluate: $$f\left ( a,b \right )=\frac{1}{a}\sum_{j=1}^{a}\cos\left ( \frac{2\pi jb}{a} \right )$$ Hence, find a function $g\left ( n \right )$, $n \in ...
2
votes
2answers
57 views

Avoiding primefactors in reducible polynomials

Take distinct pairs $(c_i,d_i) \in \mathbb Z^2$, the entries being coprime. Put $f(x) = \prod_{i=1}^k (c_i x + d_i)$. Let $\mathbb P$ denote the rational prime numbers. Which conditions (if any) need ...
2
votes
0answers
171 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
6
votes
1answer
76 views

Prove that the non-trivial root of $\sum_{k=1}^{2n} p_kx^k=0$ tends to $-1$

I looked at $$ \sum_{k=1}^{2n} p_kx^k=0, $$ where $p_k$ is the $k$th prime. I found that, next to the trivial root $x_0=0$, there is only one more root $x_n$ that tends towards $-1$, when $n$ ...
1
vote
3answers
470 views

Prime generating functions

I'm studying prime numbers at school and I've seen some functions that generate mostly prime numbers. I'm talking about : $$\text{Euler's polynomial : } n^2+n+41$$ $$\text{Legendre's polynomial : } ...
1
vote
2answers
117 views

Is polynomial $1+x+x^2+\cdots+x^{p-1}$ irreducible? [duplicate]

Let $p$ a prime number, is the polynomial $$1+x+x^2+\cdots+x^{p-1}$$ irreducible in $\mathbb{Z}[x]$ ? Thanks in advance.
4
votes
2answers
112 views

How often must an irreducible polynomial take a prime value?

Suppose $f(x)$ is an irreducible polynomial over $\mathbb Z$ of degree $n$. Is it always the case that there exist distinct $x_1,\ldots,x_{2n+1}\in \mathbb Z$ such that $f(x_1),\ldots,f(x_{2n+1})$ are ...
2
votes
2answers
165 views

Proof of lack of pure prime producing polynomials.

Now I have heard this (correct me if I am wrong) that for every polynomial, there is some positive integer for which it is composite. What is the most elementary proof of this.
8
votes
2answers
102 views

$R$ with an upper bound for degrees of irreducibles in $R[x]$

One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as If $f\in\mathbb{R}[x]$ has degree larger ...
3
votes
3answers
149 views

Cyclotomic Polynomial of a Prime

I have this question on a homework sheet: Claim:$$\Phi_{p}(x)=1+x+x^2+...+x^{p-1}\space$$ for $p$ prime. which was followed by the claim that $\Phi_{p^n}(x)=\Phi_p(x^{p^{n-1}})$ which I have ...
0
votes
1answer
203 views

Showing $f(x)$ is constant.

Let $f(x)=a_nx^n+...a_1x+a_0$ is an integer polynomial with $a_n>0,n\not=1$. $f(p)$ is prime for every $p$, where $p$ is prime. How to show $f(x)$ is constant, or not?
1
vote
0answers
126 views

Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
5
votes
2answers
287 views

Polynomials representing primes

Suppose over $\mathbb{Z}$ we are given an irreducible polynomial $p(x)$. Can we say that $p(x)$ at least represents a prime as $x$ runs through integers? Thanks in advance
0
votes
1answer
140 views

Value $\Phi_n(1)$ of the cyclotomic polynomial at x=1 [duplicate]

Possible Duplicate: Value of cyclotomic polynomial evaluated at 1 I have to show $\Phi_n(1)=1$ for $n\neq p^k$ with $p$ is prime. (I already proved to easy part $\Phi_n(1)=p$ for $n=p^k$) ...
-4
votes
5answers
379 views

Is $n^2+n+41$ prime for all whole numbers $n$?

Is $n^2+n+41$ prime for all whole numbers $n$? Furthermore, how can we prove/disprove this? Oh, sorry, I meant 41...
1
vote
1answer
71 views

Canonical form for primes in polynomial progressions

The primes of the form $4n^2-8n-9$ are just those of the form $n^2-13.$ Various substitutions can change this into many other forms. Is there a canonical choice for the polynomial, such that ...
6
votes
1answer
515 views

Are the primes found as a subset in this sequence $a_n$?

Below is a introduction that contains some background to my question. The question is found at the bottom. By calculating the eigenvalues of the matrix defined by the recurrence: $\displaystyle ...
5
votes
4answers
362 views

Show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$

Let $n$ be an integer and show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$, and is composite for all other integer values of $n$.
4
votes
1answer
102 views

Partitioning polynomials in $\mathbb{Z}[x,y]$ by the primes they represent

Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For ...
6
votes
3answers
250 views

What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$?

Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value ...
4
votes
2answers
828 views

The Prime Polynomial : Generating Prime Numbers

First of all, i'll confess i'm no math geek. I'm from Stackoverflow, but this question seemed more apt here, so i decided to ask you guys :) Now, i know noone has discovered (or ever will) a ...