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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?

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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
```

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