Tagged Questions
2
votes
1answer
67 views
Factorization of the trinomial $x^{2n}+Dx^n+1$?
The following trinomials will factor for any $a$,
$$1+a(-3+a^2)x^3+x^6 = (1+ax+x^2)(1-ax-x^2+a^2x^2-ax^3+x^4)\tag{1}$$
and similarly for,
$$1+a(5-5a^2+a^4)x^5+x^{10}\tag{2}$$
...
6
votes
6answers
182 views
Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
1
vote
0answers
36 views
Upper bound on degree of coefficients required to write polynomials as a linear combination of $f_1,…,f_n$
All polynomials will be elements of $\mathbb{Q}[x]$. Suppose $f_1,...,f_n$ are polynomials of degree at most $d$ which are coprime. What is a (hopefully sharp) upper bound on the degree of ...
0
votes
1answer
20 views
Showing a polynomial $f\in\mathbb Q[x]$ is irreducible if it has rational coefficients?
I'm trying to figure out how I can do this for some arbitrary function. Say I find a monic associate of $f$ that we'll call $f_1(x)$. If I then apply Eisenstein's Criterion or Descartes' Rational Root ...
2
votes
2answers
79 views
Euclidean Algorithm for GCD of polynomials
I am struggling to use the Euclidean algorithm for polynomials.
Given something like $$GCD(x^5+1, x^3+1)$$
I can easily use it as follows:
$$x^5+1 = x^2(x^3+1) -x^2 +1 \\
x^3+1 = -x(-x^2+1) + x +1 ...
4
votes
2answers
109 views
How to find the roots of $f(x)=x^{2}+2x+2$ in $\mathbb{Z}_{3}$ ? in $\mathbb{Z}_{5}$ ? in $\mathbb{R}$?
Normally I just guess a root and then smash one out in high degree functions, or complete squares or any other number of mathemagical tricks, but my textbook has decided to break numbers on me and I ...
0
votes
4answers
84 views
Polynomial division in $\mathbb{Z}_n[x]$
For which value of $n$ is $x^3-x$ divisible by $2x-1$ modulo $n$?
5
votes
5answers
113 views
Proving $n+3 \mid 3n^3-11n+48$
I'm really stuck while I'm trying to prove this statement:
$\forall n \in \mathbb{N},\quad (n+3) \mid (3n^3-11n+48)$.
I couldn't even how to start.
1
vote
1answer
76 views
Prove that there is no polynomial with integer coefficients such that $p(a)=b,\,p(b)=c\,p(c)=a$ for distinct integers $a,b,c$
Our teacher gave us this question but I am very stuck. I drew graphs to see why it cant be true but I didnt find anything. I see that if $p$ existed then: $$p(\cdots p(a))=a,\;p(\cdots ...
3
votes
4answers
50 views
Representation of functions between residue classes of $\mathbb{Z}/m\mathbb{Z}$ as polynomials
I am studying polynomial congruences, and there is a result that states that for any function $F: \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{Z}/p\mathbb{Z}$ (where $p$ is prime), there is a polynomial ...
2
votes
1answer
58 views
Conclusion about Zeros of a polynomial ,when sum of it's coefficients is zero
I have a polynomial of the form:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!x^{2k-2m}}{m!(2k-m)!(2k-2m+1)!}$$
or identically:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!(x^{2})^{k-m}}{m!(2k-m)!(2k-2m+1)!}$$
...
0
votes
2answers
93 views
Root of a polynomial with rational coefficients
I am currently learning about Direct Proofs. I am struggling trying to find a starting point to prove the Statement: For all real numbers $c$, if $c$ is a root of a polynomial with rational ...
1
vote
1answer
40 views
Polynomial equations over $\mathbb{Z}_m$ where m is not a power of a prime
When I have a polynomial equation over $\mathbb{Z}_m$ and $m=p_1^{l_1}\cdot \ldots \cdot p_k^{l_k} $
First for every $\,p_i^{l_i}\,$ I solve using Hensel Lifting $\,f(x)\equiv 0\,(mod\,p_i^{l_i})$
For ...
4
votes
0answers
63 views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
38
votes
4answers
2k views
Prove every odd integer is the difference of two squares
I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!
8
votes
1answer
157 views
Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?
Consider a modified version of Collatz sequence:
$C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$
Let $F_n$ be the ...
1
vote
1answer
42 views
In $\mathbb R$ is everything a unit and associated with each other?
Let $R$ be an integral domain with identity.
A unit of $R$ is an element $u \in R$ which divides 1.
Does this mean every element in $\mathbb R$ (real numbers) is a unit since every element divides ...
1
vote
2answers
73 views
Another GCD question: $(x^2-x+1)$ and $(x^3-x^2+2)$
So according to WolframAlpha the answer should be $1$ (And by inspection it certainly looks like it should be $1$.
My current working using Euclidean algorithm and polynomial long division
...
1
vote
1answer
50 views
Find monic gcd($x^4+x^3+x+1$, $x^6+x^5+x^4+x+1$) in $\mathbb Z_{2}$
My working so far using the Euclidean algorithm and polynomial long division (which I won't fully show here)
$x^6+x^5+x^4x+1$ = $(x^2+1) \times (x^4+x^3+x+1) + (-2x^3-x^2)$
and $(-2x^3-x^2) \equiv ...
2
votes
2answers
125 views
Solving polynomials in $\mathbb{Q}[X]$ exactly
I wanted to write an equation solver for rational polynomials in one variable $X$. However, such solutions do not need to be in $\mathbb{Q}$. What I wanted was to display solutions "lossless", i.e. ...
2
votes
2answers
149 views
Test for an Integer Solution
This came up an a training piece for the Putnam Competition and also in Ireland and Rosen.
The question posed was basically:
Let $p(x)$ be a polynomial with integer coefficients satisfying that ...
2
votes
1answer
95 views
Farey sequences for polynomials?
Does a notion of Farey sequence (or something equivalent) exist for polynomials over finite fields?
32
votes
1answer
994 views
Decomposing polynomials with integer coefficients
Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
3
votes
2answers
350 views
find the GCD of two polynomials of equal degree
Q: Find the GCD of the polynomials $x^3 + 4$ and $x^3 + 4x^2 + x +3$ modulo $7$.
The problem I found is that this question doesn't elaborate on modulo or which polynomial to use if both are of ...
0
votes
0answers
55 views
Given that $x_{k+1}=P(x_k)$ show that $P(x)=x+1$ [duplicate]
Possible Duplicate:
Iterated polynomial problem
Let $P(x)$ be a polynomial with integer coefficients. For each positive integer $P(n)>n$. Consider the sequence defined by $x_1=1$ and ...
13
votes
1answer
288 views
Roots with equal fractional parts
Question. ¿Does there exist an integer $n>1$ such that there exist positive integers $a,b$ such that $\{\sqrt[n]{a}\}=\{\sqrt[n]{b}\},a\neq b$ and $a$ and $b$ aren't perfect n-th powers? ( $\{x\}$ ...
2
votes
1answer
102 views
Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$?
Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$?
The values are numbers such that each prime factor of the form $4k+3$ occurs ...
2
votes
2answers
88 views
Congruence question
Hi I would like a hint with the following congruence question.
$$1+x^{1}+x^{2}+\cdots +x^{6}\equiv 0\mod{29}$$
Is there a formula I should be looking for to group the left hand side?
-4
votes
5answers
251 views
Is $n^2+n+41$ prime for all whole numbers $n$?
Is $n^2+n+41$ prime for all whole numbers $n$?
Furthermore, how can we prove/disprove this?
Oh, sorry, I meant 41...
2
votes
0answers
148 views
Extension based on similarity between rational functions and rational numbers
After realizing the similarity between rational functions as to polynomial functions and rational numbers as to integers, and the similarity of algebraic function to algebraic numbers, I am tempted to ...
0
votes
0answers
46 views
For any prime $p \equiv 1\pmod{5}$ what integers $\{a_0, \dots, a_4\}$ satisfy $(\sum_{i=0}^{4}{a_ig^i})(\sum_{i=0}^{4}{a_ig^{-i}})=p^2$?
For any prime $p \equiv 1 \pmod{5}$ do there exist 5 integers $\{a_0, \dots, a_4\}$, each of absolute value less than $p$, satisfying $\sum_{i=0}^{4}{a_i}=p$, ...
2
votes
2answers
80 views
Find and prove an upper bound on the number of intersections on two distinct polynomials
Find and prove an upper bound on the number of times that two distinct polynomials of degree $d$ can intersect.
What if the polynomials' degrees differ?
My attempt:
let $p(x)$ and $q(x)$ be two ...
3
votes
1answer
170 views
Proving a polynomial $f(x)$ composite for infinitely many $x$
Let $f(x)=a_0+a_1x+ \ldots +a_nx^n$ be a polynomial with integer
coefficients, where $a_n>0$ and $n \ge 1$. Prove that $f(x)$ is
composite for infinitely many integers $x$.
I can easily ...
3
votes
1answer
145 views
How to find such $a$ that $x^2+ax+a$ is composite for all $x$?
In my previous questions it is shown that $f(x)=x^2+ax+a$ , where $a\in\mathbf{Z^+}$\ $\left \{ 4 \right \}$ is irreducible and that gcd$(f(1),f(2),f(3).....)=1$
So, according to Bunyakovsky ...
0
votes
2answers
63 views
$\gcd(P(a),Q(a),R(a),S(a),T(a))=1$ for any particular value of $a$?
Let's define five binomials as :
$P(a)=2a+1$
$Q(a)=3a+4$
$R(a)=4a+9$
$S(a)=5a+16$
$T(a)=6a+25$
How to prove that :
$\gcd(P(a),Q(a),R(a),S(a),T(a))=1$
for any particular value of $a$ , ...
2
votes
3answers
163 views
Divisors of all values of polynomial over $\Bbb Z\,$ (fixed divisors)
From Fundamentals of Number Theory by LeVeque, section 3.1, prob. 1
Let $f(x) = a_0x^n + \cdots + a_n$ be a polynomial over Z. Show that if $r$ consecutive values of $f$ (i.e., values for consecutive ...
5
votes
4answers
301 views
Show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$
Let $n$ be an integer and show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$, and is composite for all other integer values of $n$.
4
votes
1answer
213 views
What is the formula for this function $f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$
I wonder if there exists a formula for this function?
$$f(x) = (x-1)(x-2)(x-3) \cdots (x-k)$$
I want to know the coefficient of each $x^i$, and the first thing I came up with was to find the expansion ...
3
votes
2answers
110 views
The equation $F(x) \equiv 0 \pmod m$ has integer solution for x
Let $F(x)=(x^2-17)(x^2-19)(x^2-323)$ and let $m$ be a positive integer. How can one show that the equation $F(x) \equiv 0 \pmod m$ has an integer solution?
