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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 23 09:57 EDT 2020. Contains 337298 sequences. (Running on oeis4.)