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.
Definition
Let [math]\displaystyle{ p }[/math] be a prime number. When each term in the sequence of Fibonacci numbers [math]\displaystyle{ F_n }[/math] is reduced modulo [math]\displaystyle{ p }[/math], the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted [math]\displaystyle{ \pi(p) }[/math]. Since [math]\displaystyle{ F_0 = 0 }[/math], it follows that p divides [math]\displaystyle{ F_{\pi(p)} }[/math]. A prime p such that p^{2} divides [math]\displaystyle{ F_{\pi(p)} }[/math] is called a Wall–Sun–Sun prime.
Equivalent definitions
If [math]\displaystyle{ \alpha(m) }[/math] denotes the rank of apparition modulo [math]\displaystyle{ m }[/math] (i.e., [math]\displaystyle{ \alpha(m) }[/math] is the smallest positive index such that [math]\displaystyle{ m }[/math] divides [math]\displaystyle{ F_{\alpha(m)} }[/math]), then a Wall–Sun–Sun prime can be equivalently defined as a prime [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ p^2 }[/math] divides [math]\displaystyle{ F_{\alpha(p)} }[/math].
For a prime p ≠ 2, 5, the rank of apparition [math]\displaystyle{ \alpha(p) }[/math] is known to divide [math]\displaystyle{ p  \left(\tfrac{p}{5}\right) }[/math], where the Legendre symbol [math]\displaystyle{ \textstyle\left(\frac{p}{5}\right) }[/math] has the values
 [math]\displaystyle{ \left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5;\\ 1 &\text{if }p \equiv \pm2 \pmod 5.\end{cases} }[/math]
This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ p^2 }[/math] divides the Fibonacci number [math]\displaystyle{ F_{p  \left(\frac{p}{5}\right)} }[/math].^{[1]}
A prime [math]\displaystyle{ p }[/math] is a Wall–Sun–Sun prime if and only if [math]\displaystyle{ \pi(p^2) = \pi(p) }[/math].
A prime [math]\displaystyle{ p }[/math] is a Wall–Sun–Sun prime if and only if [math]\displaystyle{ L_p \equiv 1 \pmod{p^2} }[/math], where [math]\displaystyle{ L_p }[/math] is the [math]\displaystyle{ p }[/math]th Lucas number.^{[2]}^{:42}
McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes.^{[3]} In particular, let [math]\displaystyle{ \epsilon = \left(\tfrac{p}{5}\right) }[/math]; then the following are equivalent:
 [math]\displaystyle{ F_{p  \epsilon} \equiv 0 \pmod{p^2} }[/math]
 [math]\displaystyle{ L_{p  \epsilon} \equiv 2\epsilon \pmod{p^4} }[/math]
 [math]\displaystyle{ L_{p  \epsilon} \equiv 2\epsilon \pmod{p^3} }[/math]
 [math]\displaystyle{ F_p \equiv \epsilon \pmod{p^2} }[/math]
 [math]\displaystyle{ L_p \equiv 1 \pmod{p^2} }[/math]
Existence
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 [math]\displaystyle{ k(p) }[/math], Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than [math]\displaystyle{ 10000 }[/math]. In 1960, he wrote:^{[4]}
The most perplexing problem we have met in this study concerns the hypothesis [math]\displaystyle{ k(p^2) \neq k(p) }[/math]. We have run a test on digital computer which shows that [math]\displaystyle{ k(p^2) \neq k(p) }[/math] for all [math]\displaystyle{ p }[/math] up to [math]\displaystyle{ 10000 }[/math]; however, we cannot prove that [math]\displaystyle{ k(p^2) = k(p) }[/math] is impossible. The question is closely related to another one, "can a number [math]\displaystyle{ x }[/math] have the same order mod [math]\displaystyle{ p }[/math] and mod [math]\displaystyle{ p^2 }[/math]?", for which rare cases give an affirmative answer (e.g., [math]\displaystyle{ x=3, p=11 }[/math]; [math]\displaystyle{ x=2, p=1093 }[/math]); hence, one might conjecture that equality may hold for some exceptional [math]\displaystyle{ p }[/math].
It has since been conjectured that there are infinitely many Wall–Sun–Sun primes.^{[5]} No Wall–Sun–Sun primes are known (As of August 2022).
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 2017.^{[8]} In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously.^{[9]} The project ended in December 2022, definitely proving that any Wall–Sun–Sun prime must exceed [math]\displaystyle{ 2^{64} }[/math] (about [math]\displaystyle{ 18\cdot 10^{18} }[/math]).^{[10]}
History
Wall–Sun–Sun primes are named after Donald Dines Wall,^{[4]}^{[11]} 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.^{[12]} 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
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}.^{[13]}
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
A prime p such that [math]\displaystyle{ F_{p  \left(\frac{p}{5}\right)} \equiv Ap \pmod{p^2} }[/math] with small A is called nearWall–Sun–Sun prime.^{[3]} NearWall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes. PrimeGrid recorded cases with A ≤ 1000.^{[14]} A dozen cases are known where A = ±1 (sequence A347565 in the OEIS).
Wall–Sun–Sun primes with discriminant D
Wall–Sun–Sun primes can be considered for the field [math]\displaystyle{ Q_{\sqrt{D}} }[/math] 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 [math]\displaystyle{ (P,Q) = (k,1) }[/math] corresponds to the kWall–Sun–Sun primes, for which Wall–Sun–Sun primes represent the special case k = 1. The kWall–Sun–Sun primes can be explicitly defined as primes p such that p^{2} divides the kFibonacci number [math]\displaystyle{ F_k(\pi_k(p)) }[/math], where F_{k}(n) = U_{n}(k, −1) is a Lucas sequence of the first kind with discriminant D = k^{2} + 4 and [math]\displaystyle{ \pi_k(p) }[/math] is the Pisano period of kFibonacci numbers modulo p.^{[15]} For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following.
 p^{2} divides [math]\displaystyle{ F_k\left(p  \left(\tfrac{D}{p}\right)\right) }[/math], where [math]\displaystyle{ \left(\tfrac{D}{p}\right) }[/math] 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 primes for k = 2, 3, ... are
k  squarefree part of D (OEIS: A013946)  kWall–Sun–Sun primes  notes 

1  5  ...  None are known. 
2  2  13, 31, 1546463, ...  
3  13  241, ...  
4  5  2, 3, ...  Since this is the second value of k for which D=5, the kWall–Sun–Sun primes include the prime factors of 2*2−1 that do not divide 5. Since k is divisible by 4, 2 is a kWall–Sun–Sun prime. 
5  29  3, 11, ...  
6  10  191, 643, 134339, 25233137, ...  
7  53  5, ...  
8  17  2, ...  Since k is divisible by 4, 2 is a kWall–Sun–Sun prime. 
9  85  3, 204520559, ...  
10  26  2683, 3967, 18587, ...  
11  5  ...  Since this is the third value of k for which D=5, the kWall–Sun–Sun primes include the prime factors of 2*3−1 that do not divide 5. 
12  37  2, 7, 89, 257, 631, ...  Since k is divisible by 4, 2 is a kWall–Sun–Sun prime. 
13  173  3, 227, 392893, ...  
14  2  3, 13, 31, 1546463, ...  Since this is the second value of k for which D=2, the kWall–Sun–Sun primes include the prime factors of 2*2−1 that do not divide 2. 
15  229  29, 4253, ...  
16  65  2, 1327, 8831, 569831, ...  Since k is divisible by 4, 2 is a kWall–Sun–Sun prime. 
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, 2 is a kWall–Sun–Sun prime. 
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, 2 is a kWall–Sun–Sun prime. 
25  629  5, 7, 2687, ...  
26  170  79, ...  
27  733  3, 1663, ...  
28  197  2, 1431615389, ...  Since k is divisible by 4, 2 is a kWall–Sun–Sun prime. 
29  5  7, ...  Since this is the fourth value of k for which D=5, the kWall–Sun–Sun primes include the prime factors of 2*4−1 that do 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
 Wieferich prime
 Wolstenholme prime
 Wilson prime
 PrimeGrid
 Fibonacci prime
 Pisano period
 Table of congruences
References
 ↑ ^{1.0} ^{1.1} A.S. Elsenhans, J. Jahnel (2010). "The Fibonacci sequence modulo p^{2}  An investigation by computer for p < 10^{14}". arXiv:1006.0824 [math.NT].
 ↑ Andrejić, V. (2006). "On Fibonacci powers". Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17 (17): 38–44. doi:10.2298/PETF0617038A. http://www.doiserbia.nb.rs/img/doi/03538893/2006/035388930617038A.pdf.
 ↑ ^{3.0} ^{3.1} ^{3.2} McIntosh, R. J.; Roettger, E. L. (2007). "A search for Fibonacci−Wieferich and Wolstenholme primes". Mathematics of Computation 76 (260): 2087–2094. doi:10.1090/S0025571807019552. Bibcode: 2007MaCom..76.2087M. http://www.ams.org/journals/mcom/200776260/S0025571807019552/S0025571807019552.pdf.
 ↑ ^{4.0} ^{4.1} 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, http://dml.cz/dmlcz/137492.
 ↑ Dorais, F. G.; Klyve, D. W. (2010). Near Wieferich primes up to 6.7 × 10^{15}. http://wwwpersonal.umich.edu/~dorais/docs/wieferich.pdf.
 ↑ Wall–Sun–Sun Prime Search project at PrimeGrid
 ↑ [1] at PrimeGrid
 ↑ Message boards : Wieferich and WallSunSun Prime Search at PrimeGrid
 ↑ Subproject status at PrimeGrid
 ↑ Crandall, R.; Dilcher, k.; Pomerance, C. (1997). "A search for Wieferich and Wilson primes". Mathematics of Computation 66 (217): 447. Bibcode: 1997MaCom..66..433C.
 ↑ Sun, ZhiHong; Sun, ZhiWei (1992), "Fibonacci numbers and Fermat's last theorem", Acta Arithmetica 60 (4): 371–388, doi:10.4064/aa604371388, http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf
 ↑ Klaška, Jiří (2008). "A search for Tribonacci–Wieferich primes". Acta Mathematica Universitatis Ostraviensis 16 (1): 15–20. http://dml.cz/dmlcz/137497.
 ↑ Reginald McLean and PrimeGrid, WW Statistics
 ↑ S. Falcon, A. Plaza (2009). "kFibonacci sequence modulo m". Chaos, Solitons & Fractals 41 (1): 497–504. doi:10.1016/j.chaos.2008.02.014. Bibcode: 2009CSF....41..497F.
Further reading
 Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29. ISBN 0387947779. https://archive.org/details/primenumberscomp00cran_805.
 Saha, Arpan; Karthik, C. S. (2011). "A Few Equivalences of Wall–Sun–Sun Prime Conjecture". arXiv:1102.1636 [math.NT].
External links
 Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
 Weisstein, Eric W.. "Wall–Sun–Sun prime". http://mathworld.wolfram.com/WallSunSunPrime.html.
 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)
Original source: https://en.wikipedia.org/wiki/Wall–Sun–Sun prime.
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