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I'd like to expand on the answer given by @ChipHurst in the comments. My hope was that one of these approximate tests would be much faster than PrimeQ; however, that was not the case. Perhaps someone can code these methods more efficiently than shown here. Fermat's Little Theorem: Prime p and any b such that GCD[b,p]=1 implies b^(p-1)=1, mod p. Use a prime ...


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This gives you the irreducible polynomials up to order n - 1 in $\mathbb Z_2[x]$ n = 5; Table[Pick @@ Transpose[({#, IrreduciblePolynomialQ[#, Modulus -> 2]} & /@ (FromDigits[#, x] & /@ Tuples[{0, 1}, i]))], {i, n}] // Column However, for degree 31 there are 2^32 == 4,294,967,296 tuples to ...



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