All Questions
Tagged with polynomials elementary-number-theory
668
questions
10
votes
1
answer
309
views
For any polynomial f(x) with integer coefficients, is it possible that the the greatest prime factor of f(x) can be arbitrary small?
Recently I found a problem about elementary number theory: Let $f(x)=x^2+x+1$,prove that there are infinite $n\in N$ so that the the greatest prime factor of $f(n)$ is less than $n^{1.1}$.
The answer ...
- 501
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
0
votes
0
answers
34
views
How can I sort the values of variables of multivariate polynomial f by increasing order of value of f
Assume that I have a multivariate polynomial over positive integers, the coefficients $a_i$ are all 1, e.g . $f(x_1,x_2,x_3,x_4)=x_3^4+x_2+x_1^2x_2x_4^5$, each monomial and each $x_i > 0$.
I have ...
3
votes
1
answer
116
views
Possible values for infinum of polynomial with integer coefficients
I'am trying to find all natural numbers $a\in\mathbb{N}$ such that there is polynomial with integer coefficients $P\in \mathbb{Z}[x]$ with
$$\inf_{x\in\mathbb{R}}\, P(x) = \sqrt{a}.$$
If $a=b^2$ for $...
- 185
1
vote
0
answers
80
views
Multivariate Interpolation and Chinese Remainder Theorem
Suppose we have the data points $(w,x,y,z) \bmod 3$:
$$
(2, 0, 1, 1) \\
(1, 0, 2, 2) \\
(2, 2, 0, 0) \\
(0, 2, 2, 2)
$$
and the data points $(w,x,y,z) \bmod 5$:
$$
(3, 0, 1, 2) \\
(3, 0, 2, 1) \\
(2, ...
- 3,341
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
4
votes
1
answer
107
views
Show that the sequence $\{a_n\}$ is periodic.
Suppose $f(x)$ and $g(x)$ are two integer polynomials with no common complex roots. For every integer $n,$ let $a_n = \gcd(f(n),g(n))$. Show that the sequence $\{a_n\}$ is periodic.
Note that $f$ and ...
- 1,837
-3
votes
1
answer
40
views
$f(x) = 24x^4 + 30x^3 + 18x^2 + 8x + 2$ find the degree modulo $12$, $6$ and $2$
I started by saying that $12 = 2^2 \times 3$ and tried to find solutions for $0 (\mod 2)$ but I think it is undefined so, I couldn't go any further. Could someone please explain?
- 3
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
2
votes
2
answers
199
views
Show that there exists no integer $n$ such that $n^3 - n + 3$ divides $n^3 + n^2 + n + 2$
My attempt:
For $n^3 - n + 3$ to divide $n^3 + n^2 + n + 2$, it should also divide $(n^3 + n^2 + n + 2) - (n^3 - n + 3) = n^2 + 2n - 1$. I did this to reduce the degree, but I don't think it helps.
- 404
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
3
votes
1
answer
65
views
Find the term that will have the larger coefficient
Which of the expressions $$(1+x^2-x^3)^{100} \textrm{or}\:\: (1-x^2+x^3)^{100}$$ has the larger coefficient of $x^{20}$ after expending abd and collecting terms.
I can easily do this question via ...
- 1,253
2
votes
4
answers
208
views
Find natural number $x,y$ satisfy $x^2+7x+4=2^y$
Find natural number $x,y$ satisfy $x^2+7x+4=2^y$
My try: I think $(x;y)=(0;2)$ is only solution. So I try prove $y\ge3$ has no solution, by $(x+1)(x+6)-2=2^y$.
So $2\mid (x+1)(x+6)$, but this is ...
- 727
0
votes
0
answers
32
views
Find the number of coefficients not divisible by k in the expansion of $(x^k+1)^n, n\ge 1,$
Find the number of coefficients not divisible by k in the expansion of $(x^k+1)^n, n\ge 1,$ where k is an arbitrary positive integer greater than 1.
I know that modulo 2, $(x^2 + 1)^{2^i} \equiv x^{2^...
- 1,166