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Prime numbers p for which the quintic polynomial x^5 - x - 1 modulo p completely factors into linear polynomials.

4

`%I #13 Dec 07 2016 11:42:50
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`%S 1973,3769,5101,7727,8161,9631,11903,14629,16903,17737,17921,18097,
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`%T 19477,20747,20759,21727,22717,23567,25037,26681,27397,27529,28279,
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`%U 29207,29959,30497,31091,31319,33289,36097,37463,39161,39671,40151,41491,42139,42487,42689,43331,44171,44221
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`%N Prime numbers p for which the quintic polynomial x^5 - x - 1 modulo p completely factors into linear polynomials.
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`%H G. C. Greubel, <a href="/A135844/b135844.txt">Table of n, a(n) for n = 1..1000</a>
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`%t a = {}; Do[poly = PolynomialMod[x^5 - x - 1, Prime[n]]; c = FactorList[poly, Modulus -> Prime[n]]; If[Sum[c[[m]][[2]], {m, 1, Length[c]}] == 6, AppendTo[a, Prime[n]]], {n, 1, 10000}]; a
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`%o (PARI) isok(n)=#factormod(x^5-x-1,n)[,2]==5;
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`%o forprime(n=2,10^6,if(isok(n),print1(n,", "))); \\ _Joerg Arndt_, Dec 07 2016
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`%Y Cf. A135842, A135843, A135845, A135846, A135847.
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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(35) and beyond from _G. C. Greubel_, Dec 06 2016
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