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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.

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`%I #11 Dec 06 2016 22:20:20
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`%S 691,8311,11731,17291,25111,34421,40531,41131,44971,47521,51341,64891,
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`%T 70111,74161,75991,76261,86441,88471,99611,106121,110251,112121,
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`%U 117671,118171,133241,139661,145451,156941,161591,161641,164051,164471,167071,172871,175631,184291,194981,199961,200171
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`%N 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.
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`%C According to class field theory each quintic polynomial is completely reducible modulo some prime number p of the form 10k+1.
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`%D S. Kobayashi & H. Nakagawa, Resolution of Solvable Quintic Equation, Math. Japonica Vol. 87, No 5 (1992), pp. 883-886.
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`%H G. C. Greubel, <a href="/A135843/b135843.txt">Table of n, a(n) for n = 1..1000</a>
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`%t 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
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`%Y Cf. A135842.
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`%K nonn
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`%O 1,1
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`%A _Artur Jasinski_, Dec 01 2007
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`%E Terms a(20) and beyond from _G. C. Greubel_, Dec 06 2016
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