A211672 Least number k such that the polynomial x^n - x^(n-1) - ... - 1 (mod k) has more than n distinct zeros.

209, 517, 3973, 1081, 1285, 2893, 13501, 38579, 105113, 4897, 12331, 262999, 18659, 131887, 129373, 109901, 149477, 1438121, 391229, 4244563, 160853, 196031, 5187263, 946679, 1312145, 507727, 870017, 1577593, 234973, 572977, 991349, 2279233, 1476029, 1451299

Offset: 2

Author

T. D. Noe, Apr 19 2012

Keywords

• nonn

Comments

This is the characteristic polynomial of the n-step Fibonacci and Lucas sequences. These terms produce the following number of distinct zeros: 4, 6, 8, 6, 8, 8, 10, 12, 15, 12, 18. The first 11 terms are semiprimes; the 12th term has 3 factors. For prime k, the polynomial can have at most n zeros.

Crossrefs

Cf. A211671 (for prime k).

Programs

Mathematica

Table[poly = x^n - Sum[x^k, {k, 0, n - 1}]; k = 1; While[cnt = 0; Do[If[Mod[poly, k] == 0, cnt++], {x, 0, k-1}]; cnt <= n, k++]; k, {n, 2, 7}]

Extensions

More terms from Jinyuan Wang, Apr 25 2025