A245526 Unique integer r with -prime(n)/2 < r <= prime(n)/2 such that L(2*n) == r (mod prime(n)), where L(k) denotes the Lucas number A000032(k).

1, 1, -2, -2, 2, -3, -7, 3, 5, -11, -15, 8, -18, -14, 3, -12, 19, -18, 25, 14, 5, 21, 11, 7, -22, 3, 43, -40, -7, -53, 54, 23, 11, -12, -7, 41, 6, -13, -66, 71, -32, 18, 94, -20, -79, 7, -88, 12, 11, -73, 3, 29, -120, 50, 10, -60, -63, 34, 94, 47, -113, 131, -18, 128, 60, 57, 79, 22, -45, -68, 100, 100, 131, -171, 56, -166, 11, -153, -174, 10

Offset

1,3

Comments

Conjecture: a(n) is always nonzero, i.e., prime(n) never divides the Lucas number L(2*n).

We have verified this for all n = 1, ..., 2*10^6.

On Jul 26 2014, Bjorn Poonen (from MIT) found a counterexample with n = 14268177. - Zhi-Wei Sun, Jul 26 2014

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(10) = -11 since L(2*10) = 15127 == -11 (mod prime(10)=29).

Mathematica

rMod[m_, n_]:=Mod[m, n, -(n-1)/2]
a[n_]:=rMod[LucasL[2n], Prime[n]]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000032, A000040, A245525.

Sequence in context: A125721 A049798 A165198 * A024682 A091228 A181056

Adjacent sequences: A245523 A245524 A245525 * A245527 A245528 A245529

Keyword

sign

Author

Zhi-Wei Sun, Jul 25 2014

Status

approved