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.

Definition: Proth number (http://en.wikipedia.org/wiki/Proth_number) is a number of the form :

<img src="/cgi-bin/mathtex.cgi?k\cdot 2^n 1

where k is an odd positive integer and n is a positive integer such that : <img src="/cgi-bin/mathtex.cgi?2^n>k

My question : If Proth number is prime number are there some other known relations in addition to <img src="/cgi-bin/mathtex.cgi?2^n>k , between exponent n and coefficient k ?

Definition: Proth number is a number of the form :

k\cdot 2^n 1

where k is an odd positive integer and n is a positive integer such that : 2^n>k

My question : If Proth number is prime number are there some other known relations in addition to 2^n>k , between exponent n and coefficient k ?