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A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

%I A106282 #18 Feb 16 2025 08:32:57
%S A106282 3,5,23,31,37,59,67,71,89,97,113,137,157,179,181,191,223,229,251,313,
%T A106282 317,331,353,367,379,383,389,433,443,449,463,467,487,509,521,577,619,
%U A106282 631,641,643,647,653,661,691,709,719,727,751,797,823,829,839,859,881
%N A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.
%C A106282 This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644.
%C A106282 Primes of the form 3x^2+2xy+4y^2 with x and y in Z. - _T. D. Noe_, May 08 2005
%H A106282 Vincenzo Librandi, <a href="/A106282/b106282.txt">Table of n, a(n) for n = 1..300</a>
%H A106282 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H A106282 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%t A106282 t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]
%o A106282 (PARI)
%o A106282 forprime(p=2,1000,if(#polrootsmod(x^3-x^2-x-1,p)==0,print1(p,", ")));
%o A106282 /* _Joerg Arndt_, Jul 19 2012 */
%Y A106282 Primes in A028952.
%Y A106282 Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (period of Lucas and Fibonacci 3-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).
%Y A106282 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K A106282 nonn
%O A106282 1,1
%A A106282 _T. D. Noe_, May 02 2005