32 questions linked to/from Prime factor of $A=14^7+14^2+1$
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### Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$ [duplicate]

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 ...
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### 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 ...
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### 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 ...
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### prove $x^2 - x + 1$ divides $x^{10} - x^7 + x^4 + ax + b$ for some $a, b$ in an arbitrary field [duplicate]

Let $F$ be an arbitrary field, I need to prove that $x^2 - x + 1$ divides $x^{10} - x^7 + x^4 + ax + b$ for some $a, b \in F$ The difficulty that I am currently facing is that since $F$ is an ...
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### 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 ...
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### 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 ...
### 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 ...