1
vote
2answers
24 views

If $p,q$ are prime, $q$ odd $p \not\equiv 1 \pmod q$, is there an integer $x$ such that $p\mid 1+x+\ldots+x^{q-1}$

Suppose $p,q$ are two distinct prime numbers, $q \geq 3$ and $p \not\equiv 1 \pmod q$. Then I have the following problem: Prove that there is no integer $x \in \mathbb{Z}$ such that ...
0
votes
1answer
44 views

How do you determine where a polynomial evaluates to a perfect square?

How do you determine where a polynomial evaluates to a perfect square? One example would be $f(x)=x^2+148x-288$. $f(68) = 14400 = 120^2$. Another one would be $f(x)=x^2+204x-88$. $f(2) = 324 = ...
1
vote
1answer
41 views

How do you prove this theorem?

The theorem I have to prove is ...
1
vote
2answers
64 views

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 ...
1
vote
2answers
84 views

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 ...
0
votes
4answers
44 views

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 ...
2
votes
5answers
2k views

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.
-2
votes
2answers
54 views

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 $r^3+r+1=0$
2
votes
1answer
59 views

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 ...
0
votes
0answers
31 views

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 ...
3
votes
2answers
51 views

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 ...
2
votes
1answer
49 views

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 ...
4
votes
1answer
51 views

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 ...
6
votes
1answer
93 views

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 ...
0
votes
1answer
32 views

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 ...
0
votes
1answer
79 views

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 ...
5
votes
0answers
77 views

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.
0
votes
2answers
33 views

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 ...
4
votes
1answer
38 views

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 $\mathbb{Z}[x]$ ?
5
votes
2answers
57 views

integrality of certain rational numbers

Let $P,Q\in\mathbb Q[X]$ be relatively prime polynomials ($X$ being an indeterminate). Assume that $Q(0)=1$ and that $P/Q$ is in $\mathbb Z[[X]]$. Does this imply that $P$ and $Q$ are in $\mathbb ...
1
vote
2answers
67 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
57 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
81 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
88 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 ...
8
votes
1answer
404 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
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
104 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
92 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
79 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
40 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 ...
1
vote
1answer
65 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
61 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
837 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.
4
votes
1answer
142 views

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 ...
3
votes
1answer
83 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
63 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
227 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
65 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
102 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
64 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
158 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
126 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
146 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
91 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
164 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
35 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
140 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
64 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
65 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 ...