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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.
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11
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
COMMENTS
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
MATHEMATICA
Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* Michael De Vlieger, Sep 08 2021 *)
PROG
(PARI) 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, ", "))
(PARI) 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
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.
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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
COMMENTS
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).
MATHEMATICA
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 *)
PROG
(PARI) forprime(p=2, 10^2, a=fibonacci(p-kronecker(p, 5))%p^2; a=a/p; print1(a, ", "))
(Sage)
A, p = 0, 0
while A <200:
p = next_prime(p)
A = (fibonacci(p-legendre_symbol(p, 5))/p)%p
Primes p such that the absolute value of the fraction A241014( A000720(p)) / p is a record low.
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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
COMMENTS
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.
PROG
(PARI) 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<rec, rec=r; print1(p, ", ")))
Fibonacci(n-J(n,5)) mod n^2, where J is the Jacobi symbol.
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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
COMMENTS
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
PROG
(Magma) [Fibonacci(n-(KroneckerSymbol(n, 5))) mod n^2: n in [1..70]]; // Vincenzo Librandi, May 16 2015
Smallest n-Wall-Sun-Sun prime.
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3
13, 241, 2, 3, 191, 5, 2, 3, 2683
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 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)
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.
L(p) modulo p^2, where p = prime(n) and L is a Lucas number ( A000032).
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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
COMMENTS
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.
LINKS
V. Andrejic, On Fibonacci powers, Publikacije Elektrotehnickog fakulteta - serija: matematika, 17 (2006), 38-44.
MATHEMATICA
Table[Mod[LucasL[Prime[n]], Prime[n]^2], {n, 60}] (* Vincenzo Librandi, Feb 09 2016 *)
PROG
(PARI) a000032(n) = fibonacci(n+1) + fibonacci(n-1)
a(n) = my(p=prime(n)); lift(Mod(a000032(p), p^2))
(Magma) [Lucas(p) mod p^2: p in PrimesUpTo(250)]; // Bruno Berselli, Feb 09 2016
2, 3, 5, 17, 251, 733, 1063, 123863, 1677209, 6336823451747417, 104868559750360787, 7665762181374748069
COMMENTS
Very near misses for Wall-Sun-Sun primes.
Fibonacci(p-J(p,5)) mod p^3, where p is the n-th prime and J is the Jacobi symbol.
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1
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
COMMENTS
Very similar to A113650 but modulo p^3.
MATHEMATICA
a[n_]:= Mod[Fibonacci[(n-JacobiSymbol[n, 5])], Power[n, 3]]; Table[a[Prime[n]], {n, 50}]
PROG
(Sage)
p = 1
while p < 200:
print(fibonacci(p-jacobi_symbol(p, 5))%pow(p, 3), end=', ')
p = next_prime(p)
(PARI) 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 (Python)
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
a(n) is the sum of the Wieferich and Wall-Sun-Sun residues of prime(n).
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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
COMMENTS
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).
PROG
(PARI) 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
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