All Questions
Tagged with polynomials elementary-number-theory
99
questions with no upvoted or accepted answers
10
votes
0
answers
267
views
$\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$ where $p_n = n$th prime.
How can I prove in general that, for all $n\geq 2$:
$$
\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1
$$
Seems to always be true:
...
8
votes
0
answers
200
views
Prove that $\Phi_{420}(69) > \Phi_{69}(420)$
Let $\Phi_n(x)$ denote the nth cyclotomic polynomial. Prove that $\Phi_{420}(69) > \Phi_{69}(420)$.
Observe that if $\phi$ denotes the Euler-phi function, \begin{align}
\phi(420) &= (2^2-2) \...
- 1,225
7
votes
0
answers
197
views
For which positive integers $m$ and $n$ do $x^m-x$ and $x^n-x$ being integers imply that $x$ is an integer
This is inspired by
Prove that $x$ is an integer if $x^4-x$ and $x^3-x$ are integers.
For which positive integers $m$ and $n$
do $x^m-x$ and $x^n-x$
being integers
imply that
$x$ is an integer.
$x$ is ...
- 108k
7
votes
0
answers
384
views
When is the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)+1$ reducible in $\mathbb{Z}[x]$?
This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$..
(A) Find all positive integers $n$ and integers $a_1,a_2,\ldots,...
- 49.8k
6
votes
0
answers
136
views
Which polynomials are "transitive" with respect to $\mathbb{Z}/r\mathbb{Z}$?
Given a polynomial $P(x) \in \mathbb{Z}[x]$ and a ring $R$, call $P$ transitive with respect to $R$ if and only if for all $r \in R$, there exists a natural number $n$ such that $P^n(0)=r,$ where $P^n$...
- 68.1k
4
votes
0
answers
243
views
Proper divisors of $P(x)$ congruent to 1 modulo $x$
Let $P(x) $ be a polynomial of degree $n\ge 4$ with integer coefficients and constant term equal to $1$. I am interested in Polynomials $P(x) $ such that for a fixed positive integer $b$, there are ...
- 234
4
votes
0
answers
65
views
Is there a way to convert a constrained double sum to unconstrained sum?
I have a sum that is of the form
$$S_{p}(x,y)=\sum_{n=1}^{p-1}\sum_{m=1}^{p-1-n} A_{n,m}(x,y),$$
where $A_{n,m}(x,y)$ is a monomial of the form $c_{n,m}x^ny^m$.
I wish to take a $p\rightarrow\infty$ ...
4
votes
0
answers
194
views
Extension of Goldbach's conjecture to polynomials
I noticed that a slightly modified version of Goldbach's conjecture seems to hold for the quadratic $x^2+1$. Specifically, I assert for any even $n\geq 4$, there exists at least one pair $p,q\in\...
- 6,014
4
votes
0
answers
87
views
Find coefficients $a,b,c$ and the roots of polynomial: $f(x)=ax^5+3x^4+bx^3+4x^2+3x+c$
Find coefficients $a,b,c$ and the roots of polynomial:
$$f(x)=ax^5+3x^4+bx^3+4x^2+3x+c$$
with integer coefficients, if it's known that $f$ has triple integer root.
(I'm not sure if that was the ...
4
votes
0
answers
70
views
Integer polynomials with $P(n)P(n+1)\in P(\mathbb Z)$
Find all polynomials $P\in\mathbb Z[x]$ such that for any positive integer $n$, the equation $P(x)=P(n)P(n+1)$ has an integer root.
I think that the only solutions are $P(x)=0$, $P(x)=1$, $P(x)=ax+b$,...
- 9,234
4
votes
0
answers
2k
views
Find all positive values for j,k,l such that j, k, l are positive integers and (j-k)|l, (k-l)|j, (l-j)|k.
Find all possible values of $j,k,l$ such that $j, k, l$ are positive integers and $(j-k)|l, (k-l)|j, (l-j)|k$.
As I understand that using divisibility properties, it is possible to come to some ...
- 41
4
votes
0
answers
157
views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
- 2,208
3
votes
0
answers
40
views
Calculating $P(x) \bmod x^{k}$ at target value by sampling $P(x)$
Motivation
The motivation behind this question is from a computational mathematics and combinatorics background. It is often convenient to express a problem as a product of polynomials so that the ...
- 294
3
votes
0
answers
90
views
On thickness of binary polynomials
OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
- 9,592
3
votes
0
answers
87
views
Showing the irreducibility over $\mathbb{Z}[X]$ of polynomials similar to the cyclotomic polynomials
This question follows this other question.
Let $y$ be a natural number, $x$ a variable and $$ f(x,y):= \frac{x^{2y}-1}{x+1}$$ and
$$ g(x,y):= \frac{f(x,y)^{2y+1}-1}{(f(x,y)-1)(xf(x,y)+1)}.$$
For a ...
- 3,716