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How can I calculate all binary irreducible polynomials of degree 31? or how i calculate all irreducible $f$ in $\mathbb Z_2[x]$?

(The irreducible polynomial in $\mathbb Z_2[x]$ and $\mathbb R$ are distinguish; for example, in $\mathbb Z_2[x]$, we have $x^2+1=(x+1)(x+1)$.)

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I'm not quite sure what good it will be to calculate them, as there are 143.522.117. See oeis.org/… –  belisarius Nov 12 '13 at 4:08
You can count them manually (and produce them, BTW) with Tr /@ Table[ Boole /@ (IrreduciblePolynomialQ[#, Modulus -> 2] & /@ (FromDigits[#, x] & /@ Tuples[{0, 1}, i])), {i, maxdegree}], but for going up with maxdegree greater than 20 you'll need patience (and available memory) –  belisarius Nov 12 '13 at 4:13
Over R none will be irreducible. Over Q would be a different matter. Over Z_2 there will be a large number of them, as @belisarius has already indicated. –  Daniel Lichtblau Nov 12 '13 at 16:15

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