Linked Questions

2
votes
4answers
884 views

Prove that $n^5+n^4+1$ is composite for $n>1.$ [duplicate]

Prove that $f(n)=n^5+n^4+1$ is composite for $n>1, n\in\mathbb{N}$. This problem appeared on a local mathematics competition, however it looks like there is no simple method to solve it. I tried ...
2
votes
4answers
77 views

Find remainder of division of $x^3$ by $x^2-x+1$ [duplicate]

I am stuck at my exam practice here. The remainder of the division of $x^3$ by $x^2-x+1$ is ..... and that of $x^{2007}$ by $x^2-x+1$ is ..... I tried the polynomial remainder theorem but I am ...
57
votes
7answers
7k views

Polynomial division: Is this trick obvious?

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+...
8
votes
7answers
270 views

Factorise $x^5+x+1$

Factorise $$x^5+x+1$$ I'm being taught that one method to factorise this expression which is $=x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1$ $=x^3(x^2+x+1)-x^2(x^2+x+1)+x^2+x+1$ =$(x^3-x^2+1)(x^2+x+1)$ My ...
8
votes
3answers
455 views

Prove $n^5+n^4+1$ is not a prime

I have to prove that for any $n>1$, the number $n^5+n^4+1$ is not a prime.With induction I have been able to show that it is true for base case $n=2$, since $n>1$.However, I cannot break down ...
10
votes
2answers
371 views

Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$

Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$. I tried with expanding $n^{2003}+1$, but I got nothing pretty not useful. I also couldn't get any improvement, let ...
7
votes
3answers
190 views

Find a prime that divides $14^7+14^2+1$

As the title says we seek to Find a prime that divides $14^7+14^2+1$ There is a caveat though. This was part of a contest for high-school students so undergraduate Number Theory tools such as ...
6
votes
3answers
534 views

Why do all primes $n>3$ satisfy $\,309\mid 20^n-13^n-7^n$

Solve the following... $309|(20^n-13^n-7^n)$ in $\mathbb{Z}^+$. I invested lotof time to it and finally went to WolframAlpha for help by typing... Solve $309k=20^n-13^n-7^n$ over the integers. It ...
1
vote
2answers
1k views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of $x^{2^{169}}+...
3
votes
5answers
101 views

Prove that $37$ is a factor of $(10^{2014} + 10^{2015} + 2)^{2016} − 1$

How can I prove that $37$ is a factor of $(10^{2014} + 10^{2015} + 2)^{2016} − 1$, probably using basic number theory?
0
votes
3answers
138 views

How can I factor the polynomial $x^8+x^4+x^3+x^2+x+1$ over $F_2$ [closed]

I read a solution in a book and here are the steps in it. but I don't understand how $g_i(x)$ is got.So how did he get it or is there some other ways to find the answer?
4
votes
2answers
147 views

Obtaining some elements of Galois field $2^q$

Consider an $n$-tuple $(\alpha_1,\alpha_2,\cdots, \alpha_n)$ where $\alpha_i$, $1\leq i \leq n$, are elements of the Galois field $GF(2^q)$. We know that the elementary symmetric polynomial $e_j$, $...
2
votes
3answers
173 views

Proof of polynomial divisibility without using complex numbers?

My question is the same as polynomial of degree n and its divisor except I want a solution that does not make use of complex numbers Problem: Find all positive integers $n$ such that $x^2+x+1\mid (...
0
votes
4answers
101 views

How many positive integers n are there such that $3\le n\le 100$ and $x^{2^n}+x+1$ is divisible by $x^2+x+1$?

How many positive integers n are there such that $3\le n\le 100$ and $x^{2^n}+x+1$ is divisible by $x^2+x+1$? Any hints are appreciated.
4
votes
2answers
69 views

Using roots of unity to factorize polynomials

Engel's Problem Solving Strategies has the following exercise (p.260 #28). Show $x^4 +x^3+x^2+x+1 \hspace{0.1cm}\big{|}\hspace{0.1cm}x^{44}+x^{33}+x^{22}+x^{11}+1$. I've been working through these ...

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