Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in


<http://www.mscand.dk/article.php?id=1472>, 


(If the link does not work try googling: `selmer trinomials`)


says that


$$
S(n) = x^n-x-1
$$
is irreducible over the the field $k= \mathbb{Q}$ of rational numbers.


Question : What is known about the possible irreducibility (or not) of the sligthly more general trinomial


$$
T(n,m) = x^n - x^m -1
$$


(with $0 < m < n$)


over the prime field


$$
k =GF(p)
$$


such that (say)


(a)
$p>2,$


(since seems there are many known results for binary polynomials)


and


(b)
$n$ goes to infinity when $p$ goes to infinity.


EDIT: Observe that something can be said about the parity of the number
of irreducible factors: Use Stickelberger's parity theorem.



Irreducibility of some trinomials modulo $p$
Based on a small number of small cases I suspect that the majority of those do factor. 


Let $r$ be a primitive root $\mod p$ then there is an $n$ such that $r^n=r+1 \mod p$. Then $x-r$ is a factor of $x^n-x-1$ in $\mathbb{Z}\_p$. On average $n$ should be about $p/2$. Of course one can use $x^{n+p-1}-x-1$ but that seems like cheating. That is just cases with a linear factor. 


For $p=19, $ $x^k-x-1$ factors for $2 \le k \le 18$ except for $k=4$ and $k=15$.


