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 A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros. 3
 2, 5, 11, 13, 31, 43, 53, 79, 83, 89, 97, 103, 109, 131, 139, 151, 197, 199, 229, 233, 239, 251, 257, 271, 283, 313, 317, 347, 359, 367, 379, 389, 433, 443, 461, 479, 487, 521, 569, 571, 577, 593, 599, 601, 617, 631, 641, 643, 647, 659, 673, 677, 719, 769, 797 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Fibonacci n-Step Number MAPLE Res:= NULL: count:= 0: p:= 0: P:= x^4 - x^3 - x^2 - x - 1: while count < 100 do p:= nextprime(p); if [msolve(P, p)] = [] then Res:= Res, p; count:= count+1; fi od: Res; # Robert Israel, Mar 13 2024 MATHEMATICA t=Table[p=Prime[n]; cnt=0; Do[If[Mod[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 A106283_gen(): # generator of terms p = 2 while True: if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0: yield p p = nextprime(p) A106283_list = list(islice(A106283_gen(), 20)) # Chai Wah Wu, Mar 14 2024 CROSSREFS Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p). Sequence in context: A215214 A221868 A220141 * A020629 A355657 A224793 Adjacent sequences: A106280 A106281 A106282 * A106284 A106285 A106286 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 July 24 09:00 EDT 2024. Contains 374575 sequences. (Running on oeis4.)