A271782 Smallest n-Wall-Sun-Sun prime.

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

Offset: 2

Author

Felix Fröhlich, Apr 18 2016

Keywords

• nonn
• more
• hard

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

• Sergio Falcon and Ángel 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

Crossrefs

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

Programs

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

Formula

a(4n) = 2.

Extensions

Edited by Max Alekseyev, Apr 25 2016