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 A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros. 2
 3, 5, 7, 11, 13, 17, 31, 37, 41, 53, 71, 79, 83, 107, 151, 157, 199, 229, 233, 239, 241, 257, 263, 277, 281, 311, 317, 331, 337, 379, 389, 409, 431, 433, 463, 467, 521, 523, 541, 547, 557, 563, 571, 577, 607, 631, 659, 677, 727, 769, 787, 809, 827, 839, 853 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Fibonacci n-Step Number MAPLE P:= x^5-x^4-x^3-x^2-x-1: select(p -> [msolve(P, p)] = [], [seq(ithprime(i), i=1..10000)]); # Robert Israel, Mar 13 2024 MATHEMATICA t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5-x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}]; Prime[Flatten[Position[t, 0]]] PROG (Python) from itertools import islice from sympy import Poly, nextprime from sympy.abc import x def A106284_gen(): # generator of terms from sympy.abc import x p = 2 while True: if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==0: yield p p = nextprime(p) A106284_list = list(islice(A106284_gen(), 20)) # Chai Wah Wu, Mar 14 2024 CROSSREFS Cf. A106278 (number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n)), A106298, A106304 (period of Lucas and Fibonacci 5-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p). Sequence in context: A024328 A032529 A154866 * A126145 A206864 A155801 Adjacent sequences: A106281 A106282 A106283 * A106285 A106286 A106287 KEYWORD nonn AUTHOR T. D. Noe, May 02 2005 EXTENSIONS Name corrected by Robert Israel, Mar 13 2024 STATUS approved

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Last modified April 23 20:33 EDT 2024. Contains 371916 sequences. (Running on oeis4.)