Theorem 1 For any given $n(n\geqslant2)$ , there exist a $m$, such that $x^n+x^m+1$ is irreducible over binary field.
Theorem 2 For any given $n(n\geqslant4)$ , there exist a $n_1,n_2,n_3$, such that $x^n+x^{n_1}+x^{n_2}+x^{n_3}+1$ is irreducible over binary field.
To some $n$(sucn as $n=8$) Theorem 1 doesn't hold any more, but to Theorem 2 it seems that it holds always if $n\geqslant4$. Now i want to prove this theoretically, i have considered it for a very long time. Who can help me! please....
There is a lot of work on irreducible trinomials. This paper says, among other things, that irreducible trinomials over the field of two elements don't exist if $n$ is a multiple of $8$. This paper mostly works over the field of three elements, but it gives references to papers that deal with the two-element field. Another paper with some relevant results and references is this one.