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Is it true that polynomials of the form :

$ f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$

where $\gcd(n+1,k+1)=1$ , $ a\in \mathbb{Z^{+}}$ , $a$ is odd number , $a>1$, and $a_1\neq 1$

are irreducible over the ring of integers $\mathbb{Z}$?

Eisenstein's criterion , Cohn's criterion , and Perron's criterion cannot be applied to the polynomials of this form.

Example :

The polynomial $x^4+x^3+x^2+3x+3$ is irreducible over the integers but none of the criteria above can be applied on this polynomial.

EDIT :

Note that general form for $f_n$ is : $f_n=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , so condition $a_1 \neq 1$ is equivalent to the condition $k \geq 1$ . Also polynomial can be rewritten into form :

$$f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}$$

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    I ran some code to see if mathematica could factor any of these over the integers for generic $a$. It seems like the statement holds for such polynomials of degree up to 150.2011-11-17
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    @AleksVlasev,so, you haven't found any counterexample ?2011-11-17
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    No, there does not seem to be a counter-example for these. I also ran the code with explicit $a$'s for $a = 2t+1$, $t = 1, 2,3,\dots,50$ and $n \leq 50$. Still nothing.2011-11-17
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    @AleksVlasev,If you want to answer the question I think that's ok to post the code as an answer..2011-11-17
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    Could you provide some context for this problem?2011-11-17
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    @AleksVlasev,it's related to this question: http://math.stackexchange.com/q/77780/156602011-11-17
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    You should really try this question over at Mathoverflow. Another way of asking this is that you want to understand whether the Galois group of $x^{n+1} +(a-1) x^{k+1} -a$ ever has more than $2$ orbits acting on the roots of that polynomial. A certain regular poster on MO is an expert on Galois groups of trinomials. http://www.math.harvard.edu/~elkies/trinomial.html2011-11-27
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    @DavidSpeyer,I will wait until open bounty period ends. Thanks for advice. I really would like to know answer to this question...2011-11-30
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    Actually Cohn's criterion applies for the $f(x)=x^4+x^3+x^2+3x+3$, try to revert the polynomial first though to $g(x)=1/x^4f(1/x)$, because then $g(5)=2281$ is a prime. Also Murty's irreducibility criterion can be used for it, since $g(3)=337$ is a prime.2018-04-28

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