A135843 Prime numbers p of the form 10k+1 for which the pentanacci quintic polynomial x^5-x^4-x^3-x^2-x-1 modulus p is factorizable into five binomials.

691, 8311, 11731, 17291, 25111, 34421, 40531, 41131, 44971, 47521, 51341, 64891, 70111, 74161, 75991, 76261, 86441, 88471, 99611, 106121, 110251, 112121, 117671, 118171, 133241, 139661, 145451, 156941, 161591, 161641, 164051, 164471, 167071, 172871, 175631, 184291, 194981, 199961, 200171

Offset

1,1

Comments

According to class field theory each quintic polynomial is completely reducible modulo 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^4-x^3-x^2-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. A135842.

Sequence in context: A231273 A156036 A029814 * A130662 A029828 A288837

Adjacent sequences: A135840 A135841 A135842 * A135844 A135845 A135846

Keyword

nonn

Author

Artur Jasinski, Dec 01 2007

Extensions

Terms a(20) and beyond from G. C. Greubel, Dec 06 2016

Status

approved