A113650 -id:A113650 - cp's OEIS frontend

A241014 Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

1, 1, 1, 3, 5, 3, -1, 3, -8, -3, -6, 13, -2, -4, 16, -25, 10, -13, 7, -16, -15, -30, 21, 5, 37, -4, 22, 24, 26, -53, 13, 64, 58, -22, -29, 60, 44, -3, 44, -43, -5, -50, 94, 31, -56, 5, -99, 3, -73, 18, 29, 5, -59, -1, 2

Offset: 1

Felix Fröhlich, Aug 13 2014

sign

a(n) is the smallest A such that p is a near-Wall-Sun-Sun prime. A gives the value of F_p-(p/5) modulo p^2 and a value of 0 would indicate a Wall-Sun-Sun prime. A244801 is similar but always gives the positive A, while this sequence gives A with the smallest absolute value.

a(1), with p=2, is technically ambiguous between 1 and -1, so a(1)=1 is by convention. Clearly this cannot happen for n>1 (where p^2 is odd). - Jeppe Stig Nielsen, Sep 09 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

Cf. A113650, A195988, A244801, A339855, A347565.

Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* Michael De Vlieger, Sep 08 2021 *)

forprime(p=2, 1e2, a=fibonacci(p-kronecker(p, 5))%p^2; if(a>p^2/2, a-=p^2); a=a/p; print1(a, ", "))

a(n)=my(p=prime(n)); centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p,5))))[1, 2]/p \\ Charles R Greathouse IV, Aug 21 2014

A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 16, 3, 15, 26, 25, 13, 39, 39, 16, 28, 10, 48, 7, 55, 58, 49, 21, 5, 37, 97, 22, 24, 26, 60, 13, 64, 58, 117, 120, 60, 44, 160, 44, 130, 174, 131, 94, 31, 141, 5, 112, 3, 154, 18, 29, 5, 182, 250, 2, 105

Offset: 1

Felix Fröhlich, Jul 06 2014

nonn

A value of 0 indicates a Wall-Sun-Sun prime. No such prime is known and if one exists it is > 4*10^16 (cf. PrimeGrid WSS statistics).

Felix Fröhlich, Table of n, a(n) for n = 1..291043
F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 x 10^15, J. Integer Seq. Volume 14, Issue 9 (2011).
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.
PrimeGrid, Wall-Sun-Sun Prime Search statistics
PrimeGrid, Welcome to the Wall-Sun-Sun prime search

Cf. A000045, A113650, A195988.

A= 0; p = 0; While[A < 200, p = NextPrime[p];  A= Mod[(Fibonacci[p-JacobiSymbol[p,5]])/p, p]; Print[A]] (* Javier Rivera Romeu, Jan 11 2022 *)

forprime(p=2, 10^2, a=fibonacci(p-kronecker(p, 5))%p^2; a=a/p; print1(a, ", "))

A, p = 0, 0
while A <200:
  p = next_prime(p)
  A = (fibonacci(p-legendre_symbol(p, 5))/p)%p
  print(A, end=", ") #Javier Rivera Romeu, Jan 08 2022

A339855 Primes p such that the absolute value of the fraction A241014(A000720(p)) / p is a record low.

Original entry on oeis.org

2, 3, 5, 17, 41, 101, 163, 223, 251, 733, 1063, 27191, 77969, 84299, 86813, 123863, 508771, 1677209, 11634179, 91978037, 443127523, 467335159, 1041968177, 2025051311, 13941800291, 24178397183, 762383958397, 766193665711, 1551559563569, 8030311150847

Offset: 1

Jeppe Stig Nielsen, Dec 19 2020

nonn

So-called near-Wall-Sun-Sun primes. Each term is "nearer" to being Wall-Sun-Sun than all smaller primes.
If any Wall-Sun-Sun primes exist, this sequence terminates at the smallest Wall-Sun-Sun prime.
If you start from p=7 (not p=2), then the sequence will start 7, 13, 17, 41, ... instead.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..49
Ulrich Fries and PrimeGrid, PRPNet findlist for project WSS
Reginald McLean and PrimeGrid, WW Statistics
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Wikipedia, Wall-Sun-Sun prime

Cf. A241014, A113650, A296240, A347565.

rec=+oo;forprime(p=2,,r=abs(centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p, 5)-1))[1, 1]))/p^2;if(r

A113649 Fibonacci(n-J(n,5)) mod n^2, where J is the Jacobi symbol.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 21, 34, 21, 55, 55, 89, 39, 37, 160, 98, 272, 293, 57, 365, 150, 101, 345, 433, 25, 665, 696, 709, 754, 440, 775, 994, 883, 1090, 765, 1241, 481, 230, 1511, 1355, 1599, 257, 1677, 805, 20, 1382, 752, 289, 2275, 1525, 1414, 821, 1484

Offset: 1

Eric W. Weisstein, Nov 03 2005

nonn

a(n) == 0 for n > 1 iff either n is a Wall-Sun-Sun prime (when n is prime) or a 'Wall-Sun-Sun pseudoprime' (when n is composite). The numbers meeting the second criterion are those composites where the congruence in A241505 is satisfied modulo n^2. No members are known from either of those two sets of numbers. - Felix Fröhlich, May 15 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Wall-Sun-Sun Prime

Cf. A113650.

[Fibonacci(n-(KroneckerSymbol(n,5))) mod n^2: n in [1..70]]; // Vincenzo Librandi, May 16 2015

a(n)=lift(Mod([1, 1; 1, 0]^(n-kronecker(n, 5)), n^2)[1, 2]) \\ Charles R Greathouse IV, Oct 31 2011

a(n) = fibonacci(n-kronecker(n,5)) % n^2 \\ Jeppe Stig Nielsen, Jul 22 2014

A271782 Smallest n-Wall-Sun-Sun prime.

Original entry on oeis.org

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

Offset: 2

Felix Fröhlich, Apr 18 2016

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)

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

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

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

a(4n) = 2.

Edited by Max Alekseyev, Apr 25 2016

A268478 L(p) modulo p^2, where p = prime(n) and L is a Lucas number (A000032).

Original entry on oeis.org

3, 4, 11, 29, 78, 14, 103, 324, 70, 204, 497, 519, 1477, 1420, 1881, 902, 1476, 3600, 3418, 2202, 5257, 317, 914, 5074, 4269, 9192, 5666, 6421, 7086, 4182, 12193, 3800, 1097, 11677, 299, 22651, 17271, 12063, 18371, 26297, 13784, 10137, 8405, 33583, 11230

Offset: 1

Felix Fröhlich, Feb 05 2016

nonn

Lemma 7 from the Andrejic paper (p. 42): Prime p is a Wall-Sun-Sun prime iff L(p) == 1 (mod p^2). Therefore, a(n) = 1 iff A113650(n) = 0.

Felix Fröhlich, Table of n, a(n) for n = 1..10000
V. Andrejic, On Fibonacci powers, Publikacije Elektrotehnickog fakulteta - serija: matematika, 17 (2006), 38-44.

Cf. A000032, A000040, A113650, A180363.

[Lucas(p) mod p^2: p in PrimesUpTo(250)]; // Bruno Berselli, Feb 09 2016

Table[Mod[LucasL[Prime[n]], Prime[n]^2], {n, 60}] (* Vincenzo Librandi, Feb 09 2016 *)

a000032(n) = fibonacci(n+1) + fibonacci(n-1)
a(n) = my(p=prime(n)); lift(Mod(a000032(p), p^2))

a(n) = A180363(n) mod A001248(n). - Michel Marcus, Feb 09 2016

A347565 Primes p such that A241014(A000720(p)) is +1 or -1.

Original entry on oeis.org

2, 3, 5, 17, 251, 733, 1063, 123863, 1677209, 6336823451747417, 104868559750360787, 7665762181374748069

Offset: 1

Jeppe Stig Nielsen, Sep 06 2021

Very near misses for Wall-Sun-Sun primes.

Cf. A241014, A113650, A296240, A339855.

A351989 Fibonacci(p-J(p,5)) mod p^3, where p is the n-th prime and J is the Jacobi symbol.

Original entry on oeis.org

2, 3, 5, 21, 55, 377, 2584, 2584, 9867, 754, 27683, 34706, 55391, 77486, 2961, 49237, 178121, 151768, 269809, 180340, 137459, 440741, 304859, 634125, 3589, 224018, 925249, 689508, 276097, 389850, 1566164, 488892, 101791, 731140, 1838362, 3406409, 31557, 2311014, 3158805, 4571698, 2914836, 3267050, 1294789, 6599056, 7246251, 159399

Offset: 1

Javier Rivera Romeu, Feb 27 2022

Very similar to A113650 but modulo p^3.

Cf. A113650.

a[n_]:= Mod[Fibonacci[(n-JacobiSymbol[n, 5])], Power[n, 3]]; Table[a[Prime[n]], {n, 50}]

a(n) = my(p=prime(n)); lift(Mod([1, 1; 1, 0]^(p-kronecker(p, 5)), p^3)[1, 2]); \\ Michel Marcus, Feb 28 2022

from sympy import prime, fibonacci
from sympy.ntheory import jacobi_symbol
def A351989(n): return fibonacci((p := prime(n))-jacobi_symbol(p,5)) % p**3 # Chai Wah Wu, Feb 28 2022

p = 1
while p < 200:
    print(fibonacci(p-jacobi_symbol(p,5))%pow(p,3), end=', ')
    p = next_prime(p)

A339639 a(n) is the sum of the Wieferich and Wall-Sun-Sun residues of prime(n).

Original entry on oeis.org

3, 6, 20, 35, 110, 78, 493, 114, 736, 783, 961, 518, 2542, 2752, 2820, 3392, 1062, 5124, 1139, 4047, 8322, 5372, 5727, 979, 9118, 19089, 8343, 3959, 10137, 16159, 3937, 10611, 15207, 20433, 32184, 17516, 19782, 37001, 15197, 23009, 40096, 50499, 27504, 26055

Offset: 1

Felix Fröhlich, Dec 11 2020

nonn

If a(n) = 0 then prime(n) is both a Wieferich prime (A001220) and a Wall-Sun-Sun (Fibonacci-Wieferich) prime.

If the first case of Fermat's last theorem fails for a prime p, that prime is both a Wieferich prime (cf. Wieferich, 1909) and a Wall-Sun-Sun prime (cf. Sun, 1992).

Zhi-Wei Sun, Fibonacci numbers and Fermat's last theorem, Acta Arithemtica, Vol. 60, No. 4 (1992), 371-388.
A. Wieferich, Zum letzten Fermat'schen Theorem, Journal für die reine und angewandte Mathematik, 136 (1909), 293-302, DOI:10.1515/crll.1909.136.293.

Cf. A001220, A113650, A196202.

a(n) = my(p=prime(n)); lift(Mod([1, 1; 1, 0]^(p-kronecker(p, 5)), p^2)[1, 2]) + lift(Mod(2, p^2)^(p-1)) - 1

a(n) = A196202(n) + A113650(n) - 1.

cp's OEIS Frontend

A241014 Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A339855 Primes p such that the absolute value of the fraction A241014(A000720(p)) / p is a record low.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A113649 Fibonacci(n-J(n,5)) mod n^2, where J is the Jacobi symbol.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI

PARI

A271782 Smallest n-Wall-Sun-Sun prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A268478 L(p) modulo p^2, where p = prime(n) and L is a Lucas number (A000032).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A347565 Primes p such that A241014(A000720(p)) is +1 or -1.

Views

Author

Keywords

Comments

Crossrefs

A351989 Fibonacci(p-J(p,5)) mod p^3, where p is the n-th prime and J is the Jacobi symbol.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica