A211671 Least prime p such that the polynomial x^n - x^(n-1) - ... - 1 (mod p) has n distinct zeros.

2, 11, 47, 137, 691, 25621, 59233, 2424511, 2607383, 78043403, 1032758989, 80051779

Offset: 1

Author

T. D. Noe, Apr 18 2012

Keywords

• nonn
• hard
• more

Comments

This is the characteristic polynomial of the n-step Fibonacci and Lucas sequences. For composite p, the polynomial can have more than n zeros! See A211672.

Examples

			For p = 11, x^2-x-1 = (x+3)(x+7) (mod p).
For p = 47, x^3-x^2-x-1 = (x+21)(x+30)(x+42) (mod p).
For p = 137, x^4-x^3-x^2-x-1 = (x+12)(x+79)(x+85)(x+97) (mod p).
		

Crossrefs

Cf. A045468 (n=2), A106279 (n=3), A106280 (n=4), A106281 (n=5).

Cf. A211672 (for composite p).

Programs

Mathematica

Table[poly = x^n - Sum[x^k, {k, 0, n - 1}]; k = 1; While[p = Prime[k]; cnt = 0; Do[If[Mod[poly, p] == 0, cnt++], {x, 0, p - 1}]; cnt < n, k++]; p, {n, 5}]

PARI

a(n)={my(P=x^n-sum(k=0, n-1, x^k) ); forprime(p=2, oo, if(#polrootsmod(P,p)==n, return(p) ) );} \\ Joerg Arndt, Apr 15 2013

Extensions

a(8)-a(10) from Joerg Arndt, Apr 15 2013

a(11)-a(12) from Jinyuan Wang, Apr 25 2025