All Questions
Tagged with polynomials elementary-number-theory
668
questions
2
votes
0
answers
43
views
Maximal divisor of pairwise differences
Let $r \in \mathbb{N}$ be fixed. Define the function:
$$f : \mathbb{Z}^r \to \mathbb{N}, \quad (a_1, ..., a_r) \mapsto \max_{n \in \mathbb{Z}} \prod_{i<j} \gcd((n-a_i), (n-a_j))$$
It is perhaps not ...
- 2,466
4
votes
2
answers
217
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prove that $x^{20}+(1-x)^{20}-20$ is square free
Prove that $x^{20}+(1-x)^{20}-20$ is square free (i.e. has no repeated roots).
Note that the claim holds iff $p'(x)$ is coprime to $p(x)$, where $p(x)=x^{20}+(1-x)^{20}-20$. $p'(x)= 20x^{19}-20(1-x)^{...
- 2,031
7
votes
1
answer
177
views
Product of differences of roots of unity
$\DeclareMathOperator{\lcm}{lcm}$
Denote by $\mu_n$ a primitive n-th root of unity. Examples suggest that
$$\prod_{0\le k\lt a \\ 0 \le l \lt b \\ k/a \neq l/b} (\mu_a^k - \mu_b^l)=\pm \lcm(a,b)^{\gcd(...
- 1,252
2
votes
0
answers
109
views
smallest monic polynomial that induces the zero function modulo $p^k$
I came across this problem while doing an assignment from my number theory class, and hope that someone can point me to a correct direction!
Define $d(n)$ to be the degree of the smallest monic ...
- 21
20
votes
2
answers
709
views
For what $n$ can we find a degree $\leq n-2$ polynomial such that $P(i) \in \{0 , 1 \}$ for $i \in [n]$, but not all identical.
For what $n$ is the following statement true:
There exists a choice of $ a_1, a_2, \ldots a_n \in \{ 0, 1 \}$, not all identical, such that there is a polynomial $F(x) \in \mathbb{R}[x]$ of degree at ...
- 70.4k
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
41
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Prove that $\prod_{a=0}^9 \prod_{b=0}^{100}\prod_{c=0}^{100}(\omega^a + z^b + z^c)$ is an integer congruent to $13\mod 101$
Let $\omega= e^{2\pi i/10}, z= e^{2\pi i/101}$. Prove that $\prod_{a=0}^9 \prod_{b=0}^{100}\prod_{c=0}^{100}(\omega^a + z^b + z^c)$ is an integer congruent to $13\mod 101$.
I'd prefer to avoid using ...
- 2,031
0
votes
0
answers
41
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Rewriting some fraction (Leudesdorf elementary number theory results)
In the following $p$ is a prime number, $M(p)$ is a multiple of $p$, $\mu$ is an odd number. In the cited article (see below), page 2 (200), what properties are used to rewrite from line 4 to line 5? ...
- 9
3
votes
2
answers
74
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Does $(x^2+xy+y^2)$ divide $(x+y)^n-x^n-y^n$ if and only if n has no prime factors less than 5
I noticed that for $n$ with prime factors greater than 5, that $xy(x+y)(x^2+xy+y^2)$ always seems to divide $(x+y)^n-x^n-y^n$. The $xy(x+y)$ factors seem fairly obvious, but I can't figure out where ...
- 305
2
votes
1
answer
202
views
Solving a degree-6 Diophantine inequality
While working on a proof, I ended up with the following Diophantine inequality of degree-6:
$$
a_1^6-2a_1^5+a_1^4 (1-4n)+8a_1^3 n+a_1^2 (4n^2+8n)-8a_1 n^2+16n^2 \ge 0 \tag 1
$$
The variable is $a_1$ ...
- 3,341
0
votes
1
answer
160
views
Modular Arithmetic For Polynomials
I'm teaching myself modular arithmetic but cannot seem to find any good resources on modular arithmetic regarding polynomials modulo $n \in \mathbb{N}$. How does one solve
$$P(x) \quad\text{mod}n$$
...
user1107557
6
votes
2
answers
277
views
Prove that $p(x)=x+1$.
Let $p(x)$ be a polynomial with integer coefficients such that $p(n) > n$ for every positive integer $n$. Define a sequence by $x_1 = 1, x_{i+1}=p(x_i)$ for $i\ge 1$. Suppose for any positive ...
- 1,837
4
votes
1
answer
281
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Prove that there exist infinitely many positive integer $n$ such that $\sqrt[d]{f(n)}\notin \mathbb{Z}$.
Let $f(x)\in \mathbb{Z}[x]$ be a polynomial of degree $m\geq 1$, and $d\in \mathbb{N}$ does not divide $m$. Prove that there exist infinitely many positive integer $n$ such that $\sqrt[d]{f(n)}\notin \...
user1086223
5
votes
2
answers
244
views
$g\in \mathbb{Q}[x]$ is a polynomial of degree $2022$. Prove there exist infinitely many rational $q$ such that $\sqrt[5]{g(q)}\notin \mathbb{Q}$.
Let $g\in \mathbb{Q}[x]$ be a polynomial of degree 2022. Prove that there exist infinitely many rational $q\in (0,1)$ such that $\sqrt[5]{g(q)}\notin \mathbb{Q}$.
I encountered the above problem in ...
user1086223
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 $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ ...
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