[princeps]

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.

[princeps]

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 ?

[princeps]

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 ?