# Tagged Questions

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### proportion of primes in a polynomial sequence

It is conjectured (Bunyakovsky) that when $P(x)$ is a polynomial from $\mathbb{Z}[X]$, irreducible, with positive leading coefficient and so that the integers $P(n)$ , $n\gt0$ do not share a common ...
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### Show that there exists no integer b such that f(b) is 1993.

We are given a polynomial $f$ with integer coefficients such that for 4 distinct integers $a_1,a_2,a_3$ and $a_4$, $f(a_1)=f(a_2)=f(a_3)=f(a_4)=1991$. Show that there exists no integer $b$ such that ...
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### The divisibility of the values of quadratic polynomials in $x$, for integer $x$

I would like to know method of finding validity of the statement by proofs. 1) $8$ does not divides $x^2 - 7$ for any integral value of $x$? 2) For any odd integer $x;$ the term $(x-1)^2$ is always ...
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### Is it true that $n^2+3n+13$ is prime for all $n\in\mathbb ℤ^+$?

Prove or disprove the statement: If $n\in\mathbb ℤ^+$, then $n^2+3n+13$ is prime. I am lost here. All I know is that $n$ is greater than or equal to one, since it is a positive integer.
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### Proof by contradiction, there is no rational number $r$ [closed]

Use a proof by contradiction to show that there is no rational number $r$. For which $r^3+r+1=0$
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### Can $2a^2+2a+2ab^2+b^2$ be written algebraically as the sum of three triangular numbers?

Let $T(n)=\tfrac{1}{2}n(n+1)$ denote the $n$th triangular number. I'm looking for an identity of the form $$2a^2+2a+2ab^2+b^2 = T(f(a,b)) + T(g(a,b)) + T(h(a,b))\tag{\star}$$ where $a,b$ are ...
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### Find all possible positive integral solution to following equation? [duplicate]

Please help me finding all possible positive integral solutions to following equation: $$2(x_1+x_2+_3+x_4+x_5+x_6) + x_7 = n$$ where $$0 < x_1,x_2,x_3,x_4,x_5,x_6,x_7$$ $$1 < N$$ $N$ is ...
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### Polynomial divisibility question

Let $f_n(a)$ be a polynomial of degree $n-1$ with integer coefficients, such that $f_n(a) > 0$ when $a > 0$, and in fact $f_n(a)$ is monotonically increasing with $a$. If there exists an ...
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### For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?

It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically. My question ...
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### Silly number theory questions I can't prove.

I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+bs$, if I am restricted to $a,b \in \mathbb{N}$? To be more specific, I ...
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### Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
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### Finding all such polynomials under a gcd condition

Find all such polynomial $f(x)\in \mathbb{Z}[x]$ such that $$\forall n\in \mathbb{N} \quad \gcd(f(n),f(2^n))=1$$ This is a problem from the Indian Team Selection Test. Can someone give me a solution ...
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### What do I do wrong with Möbius method of inversion?

I use the Möbius inversion with polynomials as e.g. in the well-known inversion formula of the cyclotomic polynomials. So I have $$p_{2n}(x)=\prod_{d|n}(2q_d(x))^{\mu(\frac{n}{d})}$$ Now I get the ...
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### Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
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### A polynomial $\ f(x)$ has integer coefficients such that $\ f(0)$ and $\ f(1)$ are both odd numbers. Prove that $\ f(x) = 0$ has no integer solutions. [duplicate]

Let there be a polynomial $\ f(x)$. It has integer coefficients such that $\ f(0)$ and $\ f(1)$ are both odd numbers. Prove that $\ f(x) = 0$ has no integer solutions.
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### Show that $x$ is an algebraic number? Where $x$ is…

Can someone help me with the following problem? Show that $x=\sqrt2+\sqrt[3]3$ is an algebraic number. By finding a polynomial with rational coefficients for which $x$ is a root of. Can someone ...
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### powers of $f(x)$ where $f(x)\in\mathbb{Q} [x]\setminus\mathbb{Z} [x]$

Let $n\geq 2$ be an integer. If $f(x)\in\mathbb{Q} [x]\setminus\mathbb{Z} [x]$ can $f(x)^n$ be in $\mathbb{Z}[x]$ ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### Fast way to find the smallest root $\mod M$ of a polynomial

Suppose you're given a polynomial of degree $d$ with integer coefficients: $$P(x) = \sum_{i=0}^{d}{a_i x^i}$$ Is there a fast way to find the smallest root modulo $M$, where $M$ is some composite ...
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### 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 ...
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### 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 ...
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### 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 $x$?
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### 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 ...
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### 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 ...
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### $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [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 $\Bbb Z[x]/(x^{n+1})$ ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Q \in \Bbb Z [X]$?
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### 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 ...
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 ...
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 $k$ ...