All Questions
15
questions
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 ...
- 786
1
vote
0
answers
550
views
Can we find an integer function satisfying these conditions?
I have the following problem:
Determine a function $f(x,y)$ defined on the positive integers and satisfying each of the following:
(1) $f(x, y)$ is a positive integer
(2) $f(2,3)=31$ and $f(4,1)=17.$
(...
- 13.4k
1
vote
0
answers
96
views
Non-positively algebraic number
Let us say that an algebraic number $\alpha \in \mathbf{R}$ is non-positively algebraic if it is the root of a monic polynomial $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ ...
- 193
3
votes
0
answers
90
views
Does there exist a positive integer $k$ and an irreducible polynomial $P$ of degree at least $2$ such that $P$ is a power of $k$ infinitely often?
Does there exist a positive integer $n\in\mathbb{Z}$ and an irreducible polynomial $P\in\mathbb{Z}[X]$ of degree at least $2$ such that there are infinitely many pairs of positive integers $(m,k)$ ...
- 8,421
1
vote
1
answer
401
views
I don't understand part of a proof of the Lindemann-Weierstrass theorem
I'm looking at this PlanetMath article on the Lindemann-Weierstrass theorem, specifically the dicussion right after equation (4). I think this is about the same as what appears below equation (5) in ...
- 193
11
votes
2
answers
462
views
If $x+y^3,x^2+y^2,x^3+y$ are all integers, are $x,y$ both integers?
Let $x,y$ be both real numbers. If $x+y^3,x^2+y^2,x^3+y$ are integers, are $x,y$ both integers?
This question begins with two real numbers while usual number theory tricks are based on the ...
- 838
2
votes
0
answers
103
views
Applying Hensel's Lemma When Leading Coefficient is Not a Unit
I'm familiar with Hensel's Lemma in the case where the polynomial under consideration is monic (or has invertible leading coefficient), but I'm trying to understand how it works in the case where the ...
- 2,726
4
votes
1
answer
87
views
Can a primitive integral polynomial represent integers prime to any number?
Let $m\in\mathbb N$ be any natural number and $f(X)=aX^2+bX+c$ be a polynomial with coefficients $a,b,c\in\mathbb Z$ such that $gcd(a,b,c)=1$.
Is there an $R\in\mathbb Z$ so that $f(R)$ is prime to $...
5
votes
1
answer
96
views
Primes of the form $ 2 x^2 + 2 xy + 3 y^2 $
Why is every prime $3,7 (\bmod 20)$ of the form
$$ 2 x^2 + 2 xy + 3 y^2 $$
I do not think that form is the norm of an abelian ring ?
How to prove this ?
- 16.4k
6
votes
2
answers
247
views
Proof for elements of $\textbf{Z}[\sqrt{3}]$ regarding the existence of the norm.
Prove that if there exists elements $z$ and $x$ of $\textbf{Z}[\sqrt{3}]$ such that $N(z)=ab$ and $N(x)=a$ where $a$ and $b$ are integers, then there exists a $y \in \textbf{Z}[\sqrt{3}]$ such that $N(...
- 2,175
4
votes
2
answers
262
views
Conditions on integral cubic polynomial with cyclic group and prime coefficients
Let $f=T^{3}+aT+b\in\mathbb{Z}[T]$ irreducible. To prove that if $f$ has cyclic
Galois group and primes coefficients $a,b$, then $a=b$,
Edit: added irreducible.
(which I didn't).
The ...
- 525
2
votes
0
answers
135
views
factoring polynomials in ring of integers modulo powerful number
I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number.
For example: $x^2 - 1$ in $\textbf Z_{8}$.
I know by tinkering around that $(x - 1)(x + 1)$...
- 21
2
votes
1
answer
118
views
Why must a zero of $f \in \mathbb{Z}_p[X_1, \dots, X_m] ~ (\text{mod } p^n)$ be simple in order to lift to $\mathbb{Z}_p$?
In chapter II, section 2.2, of J-P. Serre's A Course in Arithmetic, we have the following theorem:
Theorem 1: Let $f \in \mathbb{Z}_p[X_1, \dots, X_m]$, $x = (x_i) \in \left( \mathbb{Z}_p \right)^m$...
- 4,408
1
vote
2
answers
428
views
Quadratic form has non-trivial zero?
For each of the following quadratic forms, determine whether the form has a non-trivial zero (we do not need to exhibit it):
$f(x, y, z) = 2x^2 + 3y^2 - 6z^2$;
$g(x, y, z) = 2x^2 + 3y^2 - 10z^2$;
$...
user268041
0
votes
2
answers
133
views
Show that $x$ is an algebraic number? Where $x$ is...
Can someone help me with the following problem? Show that $x=\sqrt2+\sqrt[3]3$ is an algebraic number. By finding a polynomial with rational coefficients for which $x$ is a root of.
Can someone ...
- 3