A271782 Smallest n-Wall-Sun-Sun prime.

13, 241, 2, 3, 191, 5, 2, 3, 2683

Offset

2,1

Comments

A prime p is a k-Wall-Sun-Sun prime iff p^2 divides F_k(pi_k(p)), where F_k(n) is the k-Fibonacci numbers, i.e., a Lucas sequence of first kind with (P,Q) = (k,-1), and pi_k(p) is the Pisano period of k-Fibonacci numbers modulo p (cf. A001175, A175181-A175185).

For prime p > 2 not dividing k^2 + 4, it is a k-Wall-Sun-Sun prime iff p^2 | F_k(p-((k^2+4)/p)), where ((k^2+4)/p) is the Kronecker symbol.

a(1) would be the smallest Wall-Sun-Sun prime whose existence is an open question.

a(12)..a(16) = 2, 3, 3, 29, 2. a(18)..a(33) = 3, 11, 2, 23, 3, 3, 2, 5, 79, 3, 2, 7, 23, 3, 2, 239. a(36)..a(38) = 2, 7, 17. a(40), a(41) = 2, 3. a(43)..a(46) = 5, 2, 3, 41. - R. J. Mathar, Apr 22 2016

a(17) = 1192625911, a(35) = 153794959, a(39) = 30132289567, a(47)..a(50) = 139703, 2, 3, 3. If they exist, a(11), a(34), a(42) are greater than 10^12. - Max Alekseyev, Apr 26 2016

Column k = 1 of table T(n, k) of k-th n-Wall-Sun-Sun prime (that table is not yet in the OEIS as a sequence). - Felix Fröhlich, Apr 25 2016

From Richard N. Smith, Jul 16 2019: (Start)

a(n) = 2 if and only if n is divisible by 4.

a(n) = 3 if and only if n == 5, 9, 13, 14, 18, 22, 23, 27, 31 (mod 36). (End)

Links

Table of n, a(n) for n=2..10.

S. Falcon, A. Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41:1 (2009), 497-504.

Richard N. Smith, Table of n, a(n) for n = 1..1024 (or 0 if unknown, searched up to 2^22)

Richard N. Smith, List of odd n-Wall-Sun-Sun primes <= 2^20 for 1<=n<=1024

Wikipedia, Wall-Sun-Sun prime

Formula

a(4n) = 2.

Prog

(PARI) A271782(k) = forprime(p=2, 10^8, if( (([0, 1; 1, k]*Mod(1, p^2))^(p-kronecker(k^2+4, p)))[1, 2]==0, return(p); ); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4

Crossrefs

Cf. A001177, A039951, A113649, A113650, A113651, A214028, A237517, A237835, A241014, A244801, A253247, A268478.

Sequence in context: A229384 A181035 A228795 * A157969 A218229 A020520

Adjacent sequences: A271779 A271780 A271781 * A271783 A271784 A271785

Keyword

nonn,more,hard

Author

Felix Fröhlich, Apr 18 2016

Extensions

Edited by Max Alekseyev, Apr 25 2016

Status

approved