$X^n + X + 1$ reducible in $\mathbb{F}_2$

Question

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?

Answer 1

Well, that was easy: $n=5$ is the smallest value of $n$ for which $x^n+x+1$ is reducible. Indeed, $x^5+x+1=(x^2+x+1)(x^3+x^2+1)$. The other values for wich $x^n+x+1$ is reducible in $\mathbf F_2[x]$ are, for $n<100$:
8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

I determined this with a little sage program:

A.<x>=PolynomialRing(GF(2))
for i in range(2,100):
    if not((x^i+x+1).is_irreducible()):
        print i