Is it true that polynomials of the form :



> 
> $ f(x)= x^n+x^{n-1}+...+x^{k+1}+ax^k+ax^{k-1}+...a$
> 
> 
> where $gcd(n+1,k+1)=1$ and $ a\in \mathbb{Z^{+}} $
> 
> 
> 


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


Neither of  [Eisenstein's criterion](http://en.wikipedia.org/wiki/Eisenstein%27s_criterion) and [Cohn's criterion](http://en.wikipedia.org/wiki/Cohn%27s_irreducibility_criterion) cannot be applied on the polynomials of this form. I have checked a lot of cases and it seems to be true.



Are these polynomials irreducible over ring Z of integers ?
Probably you want $a \ne 1$.


I suppose no, here are some examples:


$$ x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1) $$
$$ x^{3} + x^{2} + x + 6 = (x + 2) \cdot (x^{2} - x + 3) $$
$$ x^{3} + x^{2} + x + 21 = (x + 3) \cdot (x^{2} - 2x + 7) $$
$$ x^{3} + x^{2} + x + 52 = (x + 4) \cdot (x^{2} - 3x + 13) $$
$$ x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1) $$
$$ x^{4} + x^{3} + x^{2} + x + 12 = (x^{2} - 2x + 3) \cdot (x^{2} + 3x + 4) $$
$$ x^{4} + x^{3} + x^{2} + 12x + 12 = (x + 2) \cdot (x^{3} - x^{2} + 3x + 6) $$
$$ x^{5} + x^{4} + x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} - x + 1) \cdot (x^{2} + x + 1) $$
$$ x^{5} + x^{4} + x^{3} + x^{2} + x + 22 = (x + 2) \cdot (x^{4} - x^{3} + 3x^{2} - 5x + 11) $$


