Intereting Posts

Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics)
Refining my knowledge of the imaginary number
Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$
What is the meaning of the double turnstile symbol ($\models$)?
Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$
Integral of rational function with higher degree in numerator
How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?
The maximum product of two numbers which together contain all non-zero digits exactly once
What is the equation of the orthogonal group (as a variety/manifold)?
Are there any memorization techniques that exist for math students?
Calculating summation on integer numbers from $-\infty$ to $\infty$
Computing gradient in cylindrical polar coordinates using metric?
Bijecting a countably infinite set $S$ and its cartesian product $S \times S$
Is it true that $\mathbb{E}+|\mathbb{E}\rvert\geq\mathbb{E}\rvert]+\mathbb{E}\rvert]$?
Algorithms for symbolic definite integration?

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?

- Discrete non archimedean valued field with infinite residue field
- Volume of first cohomology of arithmetic complex
- Elementary solution to the Mordell equation $y^2=x^3+9$?
- Integral Basis for Cubic Fields
- Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.
- What is a maximal abelian extension of a number field and what does its Galois group look like?
- Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?
- Roots of an irreducible polynomial in a finite field
- What are some strong algebraic number theory PhD programs?
- How many points does the surface $\mathbb{H}$ defined with the stated expression contain in $\mathbb{F}^5_{p^2}$?

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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