All Questions
Tagged with polynomials elementary-number-theory
668
questions
-6
votes
0
answers
33
views
How is this rough draft of a proof that $(x^2+10x-3=0)∉Qred? [closed]
I wanted to prove that the above quadratic is irreducible over Q without using common methods such as quadratic formula, discriminate, completing the square, rational root theorem, Eisensteins ...
5
votes
1
answer
82
views
Given nonzero $p(x)\in\mathbb Z[x]$. Are there infinitely many integers $n$ such that $p(n)\mid n!$ is satisfied?
This problem comes from a famous exercise in elementary number theory:
Prove that there are infinitely many $n\in\mathbb Z_+$ such that $n^2+1\mid n!$.
I know a lot of ways to do this. A fairly easy ...
- 7,193
19
votes
2
answers
766
views
Is the smallest root of a polynomial always complex if the coefficients is the sequence of prime numbers?
The smallest root of a polynomial is defined as the root which has the smallest absolute value. Consider the polynomial $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$. I observed that if $a_n$, is the ...
- 951
1
vote
1
answer
36
views
Factoring integer polynomials by factoring their values then interpolating
Let $f\in\mathbb Z[x]$ be a non-constant integer polynomial, and let $n$ be a positive integer which is at least $\deg f$. Choose positive integers $a_1<\cdots<a_n$ such that $f(a_i)\neq 0$ for ...
- 275k
2
votes
0
answers
245
views
Applications of the Hermite's criterion?
I found this statement on permutation polynomials and I was wondering in which domain we can find applications and what is its aim.
Here is the criterion : «If $q=p^n$ with $p$ a prime number then $f\...
- 3,330
0
votes
3
answers
103
views
Does there exists an integer-coefficient polynomial that extracts the highest digit of an integer in base p? [closed]
Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$?
The ...
- 135k
3
votes
1
answer
149
views
Find non-zero polynomials $f(n)$ and $g(n)$ with integer coefficients such that $(n^2+3)f(n)^2+1=g(n)^2$
Find non-zero polynomials $f(n)$ and $g(n)$ with integer coefficients such that $(n^2+3)f(n)^2+1=g(n)^2$.
This problem comes from Pell's equation, which states that for $\forall m\in\mathbb{N}^*$, ...
- 275k
-4
votes
1
answer
70
views
Does this proof of $f(X)\not\equiv0\pmod{n}$ (for this specific $f$) imply there is an integer $k$ such that $f(k)\not\equiv0\pmod{n}$? [closed]
A generalization of the following statement is proven in "Primes is in P":
$(X+1)^n-X^n-1\equiv0\pmod{n}$ iff $n$ is prime.
Their proof of the only if part goes something like: if $q^k\Vert ...
- 133
3
votes
2
answers
147
views
A conjecture about the sum of Faulhaber polynomials mod 2.
Context: In this question I asked about "Finding a polynomial function for alternating $m$ numbers of odds and even numbers in sequences" I noticed that all Faulhaber polynomials are ...
- 7,929
8
votes
1
answer
169
views
Finding a polynomial function for alternating m numbers of odds and even numbers in sequences
Context: The function $(-1)^n$ alternates between $1,-1$ because $n$ alternates between odd and even, I was trying to find a function $f(n)$ that will give $m$ positives then $m$ negatives and so on ...
- 6,620
2
votes
1
answer
68
views
Polynomial Divisibility by Factorial Plus One [closed]
Find all polynomials with integer coefficients $f$ such that $f(p) \mid (p-1)! + 1$ for all primes $p$.
Using Wilson’s theorem we find that $f(p)=p;-p$ satisfy the problem. I also thought about using ...
- 275k
1
vote
0
answers
157
views
Prove $a^n+\frac{1}{a^n}$ is rational if $a+\frac{1}{a}$ is rational [duplicate]
Assume
$$a+\frac{1}{a}$$
is a rational number.
Prove that $a^n+\frac{1}{a^n}$ is a rational number.
I can easily prove it using induction. I would like to know if is it possible to prove it without ...
- 275k
3
votes
2
answers
172
views
Sufficient condition for a polynomial $f$ over $\mathbb Q$ to be linear is $f(\mathbb Q)=\mathbb Q$.
I am reading a book (Polynomial Mappings by W. Narkiewicz) on invariance of polynomial maps where I found the following result along with a proof:
Theorem: Let $f\in\mathbb Q[x]$ be a polynomial such ...
- 3,100
1
vote
2
answers
194
views
Find all prime numbers that divide 2 polynomials [duplicate]
I am trying to pass some time during the COVID-19 era. I was going through my mails and found a problem. A friend of mine said her daughter had this problem in some math contest about 2-3 years ago ...
- 275k
3
votes
2
answers
2k
views
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
I got a question to show that :
If $p$ is prime number, then
$$x^p - x \equiv x(x-1)(x-2)(x-3)\cdots (x -(p-1))\,\,\text{(mod }\,p\text{)}$$
Now I got 2 steps to show that the two polynomials ...
- 275k