All Questions
43
questions
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
0
votes
0
answers
46
views
Prove that the order of 2 modulo 2n+1 is odd
Let $n$ be a positive integer such that there exists a polynomial $f(x) \in \mathbb{F}_2[x]$ of degree n such that $f(x)\cdot x^n f(1/x) = 1+x+\cdots + x^{2n}\in \mathbb{F}_2[x]$. Prove that the order ...
- 1,225
1
vote
0
answers
133
views
$\lfloor a^m\rfloor \equiv -1\mod n $ for $a = n+\sqrt{n^2 - n}$
Let $n \ge 2$ be an integer. Let $a = n+\sqrt{n^2 - n}$. Prove that for any positive integer m, we have $\lfloor a^m\rfloor \equiv -1\mod n$.
Suppose $(x-a)(x-c) = x^2 + ux + v \in \mathbb{Z}[x]$ (...
- 869
3
votes
2
answers
421
views
Polynomial system of equations over integers
I want to solve the system of equations:
$$\begin{cases}
x^4+4y^3+6x^2+4y = -137 \\
y^4+4x^3+6y^2+4x = 472
\end{cases}
$$
$x, y \in \Bbb{Z}$.
It most definitely amounts to messing around with algebra ...
- 1,488
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
41
views
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
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
views
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
2
votes
1
answer
127
views
Show that the product of the $2^{2019}$ numbers of the form $\pm 1\pm \sqrt{2}\pm\cdots \pm \sqrt{2019}$ is the square of an integer.
Show that the product of the $2^{2019}$ numbers of the form $\pm 1\pm \sqrt{2}\pm\cdots \pm \sqrt{2019}$ is the square of an integer.
I'm aware very similar problems were asked before (e.g. here and ...
- 2,031
1
vote
2
answers
275
views
Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$
Find all polynomials $P(x)$ so that $P(x)(x+1)=(x-10)P(x+1)$.
I'm looking for a general solution to the above problem. For instance, say I was trying to find all polynomials $P$ satisfying $(x+1) P(x)...
- 2,031
4
votes
2
answers
153
views
determine if the equation $x^n+y^n+z^n+w^n=u^{n+1}$ has infinitely many solutions in distinct integers
Let $n\ge 1$. Determine if the equation $x^n+y^n+z^n+w^n=u^{n+1}$ has infinitely many solutions in distinct integers. If so, determine if there are two solutions $(x_i,y_i,z_i,w_i,u_i)$ for $i=1,2$ so ...
- 1,166
5
votes
1
answer
126
views
show that n is a power of 2 given it satisfies a combinatorial property
Let $\{a_1,\cdots, a_n\}$ and $\{b_1,\cdots, b_n\}$ be two distinct sets of positive integers such that any integer can be written as $a_i+a_j$ with $i\neq j$ in exactly as many ways as it can be ...
- 1,837
1
vote
0
answers
209
views
find the smallest integer k so that for all quadratic polynomials P with integer coefficients, one of $P(1),\cdots, P(k)$ has a 0 in base 2
Find the smallest integer k so that for all quadratic polynomials P with integer coefficients, one of $P(1),\cdots, P(k)$ has a 0 in base 2 (obviously leading zeroes don't count).
Let $\mathcal{P}$ ...
- 2,031
1
vote
1
answer
50
views
find the largest possible number of elements of a set of positive integers satisfying two number properties
A problem and solution to a past contest problem are shown below. I was wondering why the claim that $\sum F^r (x_1,\cdots, x_{r^2 + 1})\equiv 0\mod p$ implies the number of solutions to (*) is ...
- 1,166