Wall–Sun–Sun prime
Named after  Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun 

Publication year  1992 
No. of known terms  0 
Conjectured no. of terms  Infinite 
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
Contents
Definition[edit]
Let be a prime number. When each term in the sequence of Fibonacci numbers is reduced modulo , the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted . Since , it follows that p divides . A prime p such that p^{2} divides is called a Wall–Sun–Sun prime.
Equivalent definitions[edit]
If denotes the rank of apparition modulo (i.e., is the smallest index such that divides ), then a Wall–Sun–Sun prime can be equivalently defined as a prime such that divides .
For a prime p ≠ 2, 5, the rank of apparition is known to divide , where the Legendre symbol has the values
This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes such that divides the Fibonacci number .^{[1]}
A prime is a Wall–Sun–Sun prime if and only if .
A prime is a Wall–Sun–Sun prime if and only if , where is the th Lucas number.^{[2]}^{:42}
McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes.^{[3]} In particular, let ; then the following are equivalent:
Existence[edit]
Unsolved problem in mathematics: Are there any Wall–Sun–Sun primes? If yes, are there an infinite number of them? (more unsolved problems in mathematics)

In a study of the Pisano period , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than . In 1960, he wrote:^{[4]}
The most perplexing problem we have met in this study concerns the hypothesis . We have run a test on digital computer which shows that for all up to ; however, we cannot prove that is impossible. The question is closely related to another one, "can a number have the same order mod and mod ?", for which rare cases give an affirmative answer (e.g., ; ); hence, one might conjecture that equality may hold for some exceptional .
It has since been conjectured that there are infinitely many Wall–Sun–Sun primes.^{[5]} No Wall–Sun–Sun primes are known as of April 2016^{[update]}.
In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×10^{14}.^{[3]} Dorais and Klyve extended this range to 9.7×10^{14} without finding such a prime.^{[6]}
In December 2011, another search was started by the PrimeGrid project^{[7]}, however it was suspended in May of 2017.^{[8]}
History[edit]
Wall–Sun–Sun primes are named after Donald Dines Wall,^{[4]}^{[9]} Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime.^{[10]} As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a potential counterexample to this centuriesold conjecture.
Generalizations[edit]
A tribonacci–Wieferich prime is a prime p satisfying h(p) = h(p^{2}), where h is the least positive integer satisfying [T_{h},T_{h+1},T_{h+2}] ≡ [T_{0}, T_{1}, T_{2}] (mod m) and T_{n} denotes the nth tribonacci number. No tribonacci–Wieferich prime exists below 10^{11}.^{[11]}
A Pell–Wieferich prime is a prime p satisfying p^{2} divides P_{p−1}, when p congruent to 1 or 7 (mod 8), or p^{2} divides P_{p+1}, when p congruent to 3 or 5 (mod 8), where P_{n} denotes the nth Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 10^{9} (sequence A238736 in the OEIS). In fact, Pell–Wieferich primes are 2Wall–Sun–Sun primes.
NearWall–Sun–Sun primes[edit]
A prime p such that with small A is called nearWall–Sun–Sun prime.^{[3]} NearWall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes.
Wall–Sun–Sun primes with discriminant D[edit]
Wall–Sun–Sun primes can be considered in the field with discriminant D. For the conventional Wall–Sun–Sun primes, D = 5. In the general case, a Lucas–Wieferich prime p associated with (P, Q) is a Wieferich prime to base Q and a Wall–Sun–Sun prime with discriminant D = P^{2} – 4Q.^{[1]} In this definition, the prime p should be odd and not divide D.
It is conjectured that for every natural number D, there are infinitely many Wall–Sun–Sun primes with discriminant D.
The case of corresponds to the kWall–Sun–Sun primes, for which Wall–Sun–Sun primes represent a special case with k = 1. The kWall–Sun–Sun primes can be explicitly defined as primes p such that p^{2} divides the kFibonacci number , where F_{k}(n) = U_{n}(k, −1) is a Lucas sequence of first kind with discriminant D = k^{2} + 4 and is the Pisano period of kFibonacci numbers modulo p.^{[12]} For a prime p ≠ 2 and not dividing D, this condition is equivalent to any of the following two:
 p^{2} divides , where is the Kronecker symbol;
 V_{p}(k, −1) ≡ k (mod p^{2}), where V_{n}(k, −1) is a Lucas sequence of the second kind.
The smallest kWall–Sun–Sun prime for k = 2, 3, ... are
k  squarefree part of D (OEIS: A013946)  kWall–Sun–Sun primes  notes 

1  5  ...  
2  2  13, 31, 1546463, ...  
3  13  241, ...  
4  5  2, 3, ...  Since this is the second time for which D=5, thus plus the prime factors of 2*2−1 which does not divide 5. Since k is divisible by 4, thus plus the prime 2. 
5  29  3, 11, ...  
6  10  191, 643, 134339, 25233137, ...  
7  53  5, ...  
8  17  2, ...  Since k is divisible by 4, thus plus the prime 2. 
9  85  3, 204520559, ...  
10  26  2683, 3967, 18587, ...  
11  5  ...  Since this is the third time for which D=5, thus plus the prime factors of 2*3−1 which does not divide 5. 
12  37  2, 7, 89, 257, 631, ...  Since k is divisible by 4, thus plus the prime 2. 
13  173  3, 227, 392893, ...  
14  2  3, 13, 31, 1546463, ...  Since this is the second time for which D=2, thus plus the prime factors of 2*2−1 which does not divide 2. 
15  229  29, 4253, ...  
16  65  2, 1327, 8831, 569831, ...  Since k is divisible by 4, thus plus the prime 2. 
17  293  1192625911, ...  
18  82  3, 5, 11, 769, 256531, 624451181, ...  
19  365  11, 233, 165083, ...  
20  101  2, 7, 19301, ...  Since k is divisible by 4, thus plus the prime 2. 
21  445  23, 31, 193, ...  
22  122  3, 281, ...  
23  533  3, 103, ...  
24  145  2, 7, 11, 17, 37, 41, 1319, ...  Since k is divisible by 4, thus plus the prime 2. 
25  629  5, 7, 2687, ...  
26  170  79, ...  
27  733  3, 1663, ...  
28  197  2, 1431615389, ...  Since k is divisible by 4, thus plus the prime 2. 
29  5  7, ...  Since this is the fourth time for which D=5, thus plus the prime factors of 2*4−1 which does not divide 5. 
30  226  23, 1277, ... 
D  Wall–Sun–Sun primes with discriminant D (checked up to 10^{9})  OEIS sequence 

1  3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes)  A065091 
2  13, 31, 1546463, ...  A238736 
3  103, 2297860813, ...  A238490 
4  3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes)  
5  ...  
6  (3), 7, 523, ...  
7  ...  
8  13, 31, 1546463, ...  
9  (3), 5, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes)  
10  191, 643, 134339, 25233137, ...  
11  ...  
12  103, 2297860813, ...  
13  241, ...  
14  6707879, 93140353, ...  
15  (3), 181, 1039, 2917, 2401457, ...  
16  3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes)  
17  ...  
18  13, 31, 1546463, ...  
19  79, 1271731, 13599893, 31352389, ...  
20  ...  
21  46179311, ...  
22  43, 73, 409, 28477, ...  
23  7, 733, ...  
24  7, 523, ...  
25  3, (5), 7, 11, 13, 17, 19, 23, 29, ... (All odd primes)  
26  2683, 3967, 18587, ...  
27  103, 2297860813, ...  
28  ...  
29  3, 11, ...  
30  ... 
See also[edit]
 Wieferich prime
 Wolstenholme prime
 Wilson prime
 PrimeGrid
 Fibonacci prime
 Pisano period
 Table of congruences
References[edit]
 ^ ^{a} ^{b} A.S. Elsenhans, J. Jahnel (2010). "The Fibonacci sequence modulo p^{2}  An investigation by computer for p < 10^{14}". arXiv:1006.0824.
 ^ Andrejić, V. (2006). "On Fibonacci powers" (PDF). Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A.
 ^ ^{a} ^{b} ^{c} McIntosh, R. J.; Roettger, E. L. (2007). "A search for Fibonacci−Wieferich and Wolstenholme primes" (PDF). Mathematics of Computation. 76 (260): 2087–2094. Bibcode:2007MaCom..76.2087M. doi:10.1090/S0025571807019552.
 ^ ^{a} ^{b} Wall, D. D. (1960), "Fibonacci Series Modulo m", American Mathematical Monthly, 67 (6): 525–532, doi:10.2307/2309169
 ^ Klaška, Jiří (2007), "Short remark on Fibonacci−Wieferich primes", Acta Mathematica Universitatis Ostraviensis, 15 (1): 21–25.
 ^ Dorais, F. G.; Klyve, D. W. (2010). "Near Wieferich primes up to 6.7 × 10^{15}" (PDF). Cite journal requires
journal=
(help)  ^ Wall–Sun–Sun Prime Search project at PrimeGrid
 ^ [1] at PrimeGrid
 ^ Crandall, R.; Dilcher, k.; Pomerance, C. (1997). "A search for Wieferich and Wilson primes". 66: 447. Cite journal requires
journal=
(help)  ^ Sun, ZhiHong; Sun, ZhiWei (1992), "Fibonacci numbers and Fermat's last theorem" (PDF), Acta Arithmetica, 60 (4): 371–388
 ^ Klaška, Jiří (2008). "A search for Tribonacci–Wieferich primes". Acta Mathematica Universitatis Ostraviensis. 16 (1): 15–20.
 ^ S. Falcon, A. Plaza (2009). "kFibonacci sequence modulo m". Chaos, Solitons & Fractals. 41 (1): 497–504. Bibcode:2009CSF....41..497F. doi:10.1016/j.chaos.2008.02.014.
Further reading[edit]
 Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29. ISBN 0387947779.
 Saha, Arpan; Karthik, C. S. (2011). "A Few Equivalences of Wall–Sun–Sun Prime Conjecture". arXiv:1102.1636.
External links[edit]
 Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
 Weisstein, Eric W. "Wall–Sun–Sun prime". MathWorld.
 Richard McIntosh, Status of the search for Wall–Sun–Sun primes (October 2003)
 OEIS sequence A000129 (Primes p that divide their Pell quotients, where the Pell quotient of p is A000129(p  (2/p))/p and (2/p) is a Jacobi symbol)