A135842 Prime numbers p of the form 10k+1 for which the quintic polynomial x^5-x-1 modulus p is factorizable into five binomials.

5101, 8161, 9631, 17921, 26681, 31091, 39161, 39671, 40151, 41491, 43331, 44171, 44221, 48541, 75821, 77951, 84391, 94531, 109391, 111521, 113891, 114661, 117511, 118081, 124121, 132241, 141241, 144511, 156371, 160231, 161771, 167381, 174481, 178951, 184321, 184511, 186871, 187091, 204301

Offset

1,1

Comments

According to class field theory each quintic polynomial is completely reducible mod some prime number p of the form 10k+1.

References

S. Kobayashi & H. Nakagawa, Resolution of Solvable Quintic Equation, Math. Japonica Vol. 87, No 5 (1992), pp. 883-886.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Mathematica

a = {}; Do[If[PrimeQ[10n + 1], poly = PolynomialMod[x^5 - x - 1, 10n + 1]; c = FactorList[poly, Modulus -> 10n + 1]; If[Sum[c[[m]][[2]], {m, 1, Length[c]}] == 6, AppendTo[a, 10n + 1]]], {n, 1, 10000}]; a

Crossrefs

Cf. A135843.

Sequence in context: A260069 A252143 A223400 * A224492 A330730 A343081

Adjacent sequences: A135839 A135840 A135841 * A135843 A135844 A135845

Keyword

nonn

Author

Artur Jasinski, Dec 01 2007

Extensions

Terms a(19) and beyond from G. C. Greubel, Dec 06 2016

Status

approved