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

(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.

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# Irreducibility of some trinomials modulo $p$

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

(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.