1
vote
2answers
44 views

Remainder of the polynomial

A polynomial function $f(x)$ with real coefficients leaves the remainder $15$ when divided by $x-3$, and the remainder $2x+1$ when divided by $(x-1)^2$. Then the remainder when $f(x)$ is divided by ...
1
vote
3answers
52 views

zeroes to polynomials in residue rings of Z

I'm supposed to find zeroes of $x^{12} -16$ in $\mathbb Z_{17}$, seems simple enough but I just can't seem to make any progress. I realize of course that we have $X^{12} = -1$ in $\mathbb Z_{17}$, ...
0
votes
2answers
61 views

Show that the equation $x^3+7x-14(n^2+1)$ has no integral root for any integer $n$.

Show that the equation $$x^3+7x-14(n^2+1)=0$$ has no integral root for any integer $n$. My work: I consider the contraposition that there are integer roots. Assume that the roots are ...
1
vote
2answers
63 views

All roots of the polynomial equation $x^4-4x^3+ax^2+bx+1=0$ are positive real numbers. Show that all the roots of the polynomial are equal.

Suppose that all roots of the polynomial equation $$x^4-4x^3+ax^2+bx+1=0$$ are positive real numbers. Show that all the roots of the polynomial are equal. My work: I assume the contraposition that ...
6
votes
0answers
250 views

IMO 1979 problem

The question is $$If\, p,q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad Prove \,that\, 1979|p.$$ So my solution went like this: ...
0
votes
0answers
48 views

Showing a Prime Integer Divides the Content of an Integer Polynomial

Setting: $f = a_0 + a_1 x + \ldots + a_n x^n \in \mathbb{Z}[x]$ $g = x^m - 1 \in \mathbb{Z}[x]$ $f \mid g$ $f^{(i)} = f(x^i) = a_0 + a_1 x^i + \ldots + a_n (x^i)^n \in \mathbb{Z}[x]$ for all $i \in ...
1
vote
1answer
76 views

A polynomial is called a Fermat's polynomial…

A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0) = ...
2
votes
1answer
87 views

Find known number of missing natural numbers

Given a set $S$ of distinct natural numbers, we know that a subset $T$ that is $S$ with at most $k$ number of elements missing. Let $M_k := \big\{m_j\big|d_j = \sum_{i\in T}i^j, j\in ...
1
vote
1answer
65 views

Integer polynomial, maximum number of consecutive integer values that it can reach.

Lets say I have an integer polynomial $P(x)$ of degree $n$ and $x_0,\dots,x_r \in \mathbb{Z}$ such that $P(x_0) = 0$ $P(x_1) = 1$ $P(x_2) = 2$ $\dots$ $P(x_r) = r$ What is the largest $r$ that I ...
0
votes
0answers
33 views

Polynomial factorization over an infinite field - is there an algorithm

In my previous questioned I asked how do I factor a polynomial, and I gave an easy example of a polynomial of degree 2. But now I have another question I need to solve. I need to factor ...
0
votes
0answers
81 views

Incongruent solutions to $f(x)$ mod p, when $x^n$ is largest poser of $x$ with coefficient not divisible by p

Show that if $p$ is a prime, $f(x)$ is a polynomial with integer coefficients and $x^n$ is the largest power of $x$ with a coefficient not divisible by $p$, then the congruence $f(x)==0$ mod $p$ has ...
1
vote
1answer
59 views

On a Congruence Relation Between Polynomials

Problem: If $ f \in \mathbb{Z}[X] $ and $ f(a) ≡ 0 \pmod n $ for some $ a \in \mathbb{Z} $, then there exists a $ g \in \mathbb{Z}[X] $ such that $ f(X) ≡ (X − a) g(X) \pmod n $. I think that ...
1
vote
1answer
43 views

Boolean algebra generated by value sets of polynomials over $\mathbb{N}$

Update For each polynomial $P \in \mathbb{N}[X]$, let $S_P = \{ P(n) \mid n \in \mathbb{N}\}$. Does the Boolean algebra generated by the subsets $S_P$ of $\mathcal{P}(\mathbb{N})$ such that $P$ is ...
2
votes
2answers
397 views

Solving Quartic Equation

Could someone please explain how to solve this : $x^4+3x^3-6x^2+16x+56=0$ - not the answer only, but a step-by-step solution.
3
votes
0answers
93 views

Fast way to find the smallest root of modular polynomial

Suppose you're given a polynomial with integer coefficients: $$ P(x) = \sum_{i=0}^{n}{a_i x^i} $$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite number with ...
3
votes
1answer
81 views

Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$

Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$. I'd like to get some pointers about how to solve this. No full ...
1
vote
0answers
24 views

Are there any simple functions which map $\mathbb Z^n\to \mathbb Z\setminus \{k\}$ for given integer $k$?

Obviously, a function could be explicitly constructed as the set of all points in $\mathbb Z^n$ and what they are mapped to such that the given integer $k$ is not in the range. I am hoping to find a ...
0
votes
2answers
61 views

Euclid for polynomials [duplicate]

I have a question bout euclid polynomials. If $C(x) =x^4−1$ and $D(x) =x^3+x^2$ How do I find a polynomials $A(x)$ and $B(x)$ such that $A(x)C(x) +B(x)D(x) =x+1$ for all $x$?
1
vote
3answers
206 views

Show $f(X)=a_nX^n+\cdots+a_1X+a_0$ has degree $n$ modulo $N$, $f(a)\equiv 0$ (mod $N$) then $f(X) \equiv (X-a)g(X) $(mod $N$)

In Niels Lauritzen, Concrete Abstract Algebra, I'm having trouble showing the following: The problem starts out like this: $f(X)=a_nX^n+\cdots+a_1X+a_0, a_i \in \mathbb Z, n \in \mathbb N$ Part ...
1
vote
1answer
58 views

Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
2
votes
2answers
96 views

$(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [duplicate]

I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$. Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ ...
0
votes
1answer
63 views

Natural number n-Divisibility

The number of natural number $n$ in the interval $[1005,2010]$ for which the polynomial $$1+x+x^2+x^3\dots +x^{(n-1)}$$ divides the polynomial $$1+x^2+x^4\dots+x^{2010}$$ is: I could realize that ...
4
votes
1answer
148 views

Does this theorem have a name?

Let P(x) be a polynomial of degree n. Let H(i) represent the number of 1's in the binary expansion of the integer i. Although reasonably easy to prove, it may seem surprising that the following ...
3
votes
1answer
124 views

On $x^3+y^3+z^3 = 1$ and a Pell equation

Given, $$(1-ac+bc)^3 + (a+c^2-ac^3)^3 + (ac^3-b-c^2)^3 = 1\tag{1}$$ where, $$a,b,c,r = 12qrt,\;\; 3(q-r)(3q+r)t,\;\; 3s^2t^2,\;\; p-18qs^3t^3$$ then $(1)$ holds true if $p,q,s,t$ satisfies, ...
5
votes
2answers
135 views

How many times can a $4^{th}$ degree polynomial be equal to a prime number?

If $f(x)$ is a $4^{th}$ degree polynomial with integer coefficients, what is the largest set ${x_1, x_2, x_3, ...x_n}$ (where $x_i$ are integers) for which $|f(x_i)|$ is a prime number? Things I ...
6
votes
2answers
87 views

When a value of a polynomial over $\mathbb Z$ is a perfect square

For which values of $x\in\mathbb{Z}$ the polynomial $16x^3-24x+9$ is a perfect square? I don't know if this question has a solution, but Wolfram Alpha says that the answer is $x=0$ (click), even if ...
3
votes
1answer
114 views

Testing polynomial equivalence

Suppose I have two polynomials, P(x) and Q(x), of the same degree and with the same leading coefficient. How can I test if the two are equivalent in the sense that there exists some $k$ with ...
0
votes
0answers
33 views

Analogue of $\omega (n)$ in polynomial ring over finite fields

Let $n \in \mathbb{N}$ and $\omega(n)$ be the number of distinct prime divisors, which divide $n$. We know that $ \omega(n)$ ~ $\frac{\log n}{\log \log n}. $ I know there is an analogue of prime ...
3
votes
1answer
116 views

Factoring polynomials of the form $1+x+\cdots +x^{p-1}$ in finite field

Suppose $p$ and $q$ primes and $p$ is odd. Then, are there nice and clever ways to factorize polynomials of the form $$f(x)=1+x+\cdots +x^{p-1}$$ in the ring $\mathbb{F}_q[x]$ ? What about the case ...
1
vote
1answer
60 views

Polynomial whose only values are squares

Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?
1
vote
0answers
59 views

Simplify $\frac{[m+n-1]!}{[m]![n]!}$ where $[k]=x^k-x^{-k}$ and $[k]!=[2][3]…[k]$.

Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
0
votes
0answers
70 views

How to find the nearest power product?

We call power products the integers of the form $x^m*y^n$ for $m$, $n$, $x$, $y \in \mathbb{N}$. Given a number $u \in \mathbb{N}$, find the closest power product. How does one solve this ...
1
vote
2answers
152 views

Proof that any polynomial with a positive leading coefficient is eventually positive?

The exact theorem I've been asked to prove is the following: Suppose $f(x)=a_n x^n + a_{n-1}x^{n-1} + ...+a_0$ is a polynomial of degree $n>0$ and suppose $a_n>0$. Then there is an integer $k$ ...
1
vote
3answers
465 views

Prime generating functions

I'm studying prime numbers at school and I've seen some functions that generate mostly prime numbers. I'm talking about : $$\text{Euler's polynomial : } n^2+n+41$$ $$\text{Legendre's polynomial : } ...
2
votes
1answer
79 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, $$1+a(5-5a^2+a^4)x^5+x^{10}\tag{2}$$ ...
7
votes
6answers
207 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
65 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
26 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
1k 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 ...
3
votes
2answers
127 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
96 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
127 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
96 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
63 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 ...
3
votes
1answer
219 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
563 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
55 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
77 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 ...
46
votes
7answers
4k 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
189 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 ...