Wall–Sun–Sun prime

Short description: A certain type of prime number conjectured to exist but of which no example has yet been found.

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 p² 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].

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.

McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes. 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:

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. 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 > 2×10¹⁴. Dorais and Klyve extended this range to 9.7×10¹⁴ without finding such a prime.

In December 2011, another search was started by the PrimeGrid project, however it was suspended in May 2017. In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. 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]).

History

Wall–Sun–Sun primes are named after Donald Dines Wall, 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. 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 centuries-old conjecture.

Generalizations

A tribonacci–Wieferich prime is a prime p satisfying h(p) = h(p²), where h is the least positive integer satisfying [T_h,T_h+1,T_h+2] ≡ [T₀, T₁, T₂] (mod m) and T_n denotes the n-th tribonacci number. No tribonacci–Wieferich prime exists below 10¹¹.

A Pell–Wieferich prime is a prime p satisfying p² divides P_p−1, when p congruent to 1 or 7 (mod 8), or p² divides P_p+1, when p congruent to 3 or 5 (mod 8), where P_n denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 10⁹ (sequence A238736 in the OEIS). In fact, Pell–Wieferich primes are 2-Wall–Sun–Sun primes.

Near-Wall–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 near-Wall–Sun–Sun prime. Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes. PrimeGrid recorded cases with |A| ≤ 1000. 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 = P² – 4Q. 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 k-Wall–Sun–Sun primes, for which Wall–Sun–Sun primes represent the special case k = 1. The k-Wall–Sun–Sun primes can be explicitly defined as primes p such that p² divides the k-Fibonacci 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 = k² + 4 and [math]\displaystyle{ \pi_k(p) }[/math] is the Pisano period of k-Fibonacci numbers modulo p. For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following.

• p² 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 p²), where V_n(k, −1) is a Lucas sequence of the second kind.

The smallest k-Wall–Sun–Sun primes for k = 2, 3, ... are

13, 241, 2, 3, 191, 5, 2, 3, 2683, ... (sequence A271782 in the OEIS)

k	square-free part of D (OEIS: A013946)	k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–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 k-Wall–Sun–Sun prime.
29	5	7, ...	Since this is the fourth value of k for which D=5, the k-Wall–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 109)	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

Further reading

• Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29. ISBN 0-387-94777-9. 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/Wall-Sun-SunPrime.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)