Are these polynomials irreducible over integers ?
Is it true that polynomials of the form :
[itex] f_n= x^n x^{n-1} \cdots x^{k 1} ax^k ax^{k-1} \cdots a [/itex]
where [itex]\gcd(n 1,k 1)=1[/itex] , [itex] a\in \mathbb{Z^{ }}[/itex] , [itex]a[/itex] is odd number , [itex]a>1[/itex], and [itex]a_1\neq 1[/itex]
are irreducible over the ring of integers [itex]\mathbb{Z}[/itex]?
Eisenstein's criterion , Cohn's criterion , and Perron's criterion cannot be applied to the polynomials of this form.
Example :
The polynomial [itex]x^4 x^3 x^2 3x 3[/itex] is irreducible over the integers but none of the criteria above can be applied on this polynomial.
Relations between coefficient and exponent of Proth prime ?
Definition: Proth number is a number of the form :
[IMG]<img src="/cgi-bin/mathtex.cgi?k\cdot 2^n 1[/IMG]
where k is an odd positive integer and n is a positive integer such that : [IMG]<img src="/cgi-bin/mathtex.cgi?2^n>k[/IMG]
My question : If Proth number is prime number are there some other known relations in addition to [IMG]<img src="/cgi-bin/mathtex.cgi?2^n>k[/IMG] , between exponent n and coefficient k ?
Relations between coefficient and exponent of Proth prime
Definition: Proth number is a number of the form :
where 
is an odd positive integer and 
is a positive integer such that : 
My question : If Proth number is prime number are there some other known relations in addition to , between exponent and coefficient ?
