Wall–Sun–Sun prime explained

Named After:	Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun
Publication Year:	1992
Terms Number:	0
Con Number:	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

p be a prime number. When each term in the sequence of Fibonacci numbers F _{n} is reduced modulo p , the result is a periodic sequence.The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p) . Since F _{0} = 0 , it follows that p divides F _{\pi(p)} . A prime p such that p² divides F _{\pi(p)} is called a Wall–Sun–Sun prime.

Equivalent definitions

If

\alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index such that m divides F _{\alpha(m)} ), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p ^{2} divides F _{\alpha(p)} .

For a prime p ≠ 2, 5, the rank of apparition

\alpha(p) is known to divide p - \left(\tfrac{p}{ 5 }\right) , where the Legendre symbol style\left( (p)/(5) \right) has the values\left( (p)/(5) \right) = \begin{cases} 1 & if p \equiv \pm 1 \pmod 5 ;\ -1 & if p \equiv \pm 2 \pmod 5 .\end{cases} This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes p such that p ^{2} divides the Fibonacci number F _{p - \left( (p)/(5) \right)} .^[1]

A prime

p is a Wall–Sun–Sun prime if and only if \pi(p ^{2)} = \pi(p) .

A prime

p is a Wall–Sun–Sun prime if and only if L _{p} \equiv 1 \pmod{p ^{2}} , where L _{p} is the p -th Lucas number.^[2]

McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes. In particular, let

\epsilon = \left(\tfrac{p}{ 5 }\right) ; then the following are equivalent:F _{p} \equiv 0 \pmod{p ^{2}} L _{p} \equiv 2 \epsilon \pmod{p ^{4}} L _{p} \equiv 2 \epsilon \pmod{p ^{3}} F _{p} \equiv \epsilon \pmod{p ^{2}} L _{p} \equiv 1 \pmod{p ^{2}}

Existence

In a study of the Pisano period

k(p) , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than 10000 . In 1960, he wrote:It has since been conjectured that there are infinitely many Wall–Sun–Sun primes.^[3]

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2.^[4] Dorais and Klyve extended this range to 9.7 without finding such a prime.^[5]

In December 2011, another search was started by the PrimeGrid project,^[6] however it was suspended in May 2017.^[7] In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously.^[8] The project ended in December 2022, definitely proving that any Wall–Sun–Sun prime must exceed

2 ^{64} (about 18 ⋅ 10 ^{18} ).^[9]

History

Wall–Sun–Sun primes are named after Donald Dines Wall,^[10] 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, 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 n-th tribonacci number. No tribonacci–Wieferich prime exists below 10¹¹.^[11]

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 . In fact, Pell–Wieferich primes are 2-Wall–Sun–Sun primes.

Near-Wall–Sun–Sun primes

A prime p such that

F _{p - \left( (p)/(5) \right)} \equiv Ap \pmod{p ^{2}} with small |A| is called near-Wall–Sun–Sun prime.^[4] Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes. PrimeGrid recorded cases with |A| ≤ 1000.^[12] A dozen cases are known where A = ±1 .

Wall–Sun–Sun primes with discriminant D

Q _{\sqrt{D} } 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.^[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

(P,Q) = (k, -1 ) 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 F _{k(\pi} _{k(p))} , where F_k(n) = U_n(k, −1) is a Lucas sequence of the first kind with discriminant D = k² + 4 and \pi _{k(p)} is the Pisano period of k-Fibonacci numbers modulo p.^[13] For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following.

• p² divides

F _{k\left(p} - \left(\tfrac{D}{p}\right)\right) , where \left(\tfrac{D}{p}\right) 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, ...

See also

• Wieferich prime
• Wolstenholme prime
• Wilson prime
• PrimeGrid
• Fibonacci prime
• Pisano period
• Table of congruences

Further reading

• Book: Prime Numbers: A Computational Perspective . limited . Richard E. . Crandall . Carl . Pomerance . Springer . 2001 . 29 . 0-387-94777-9 .
• Saha . Arpan . Karthik . C. S. . A Few Equivalences of Wall–Sun–Sun Prime Conjecture . 2011 . 1102.1636 . math.NT .

External links

• Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
  • Richard McIntosh, Status of the search for Wall–Sun–Sun primes (October 2003)

Notes and References

1. A.-S. Elsenhans, J. Jahnel . The Fibonacci sequence modulo p² -- An investigation by computer for p < 10¹⁴ . 1006.0824 . 2010. math.NT .
2. Andrejić . V. . On Fibonacci powers . Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. . 17 . 17 . 38–44 . 2006 . 10.2298/PETF0617038A. 41226139 .
3. .
4. R. J. . McIntosh . E. L. . Roettger . A search for Fibonacci−Wieferich and Wolstenholme primes . . 76 . 260 . 2007 . 2087–2094 . 10.1090/S0025-5718-07-01955-2. 2007MaCom..76.2087M . free .
5. Web site: Dorais . F. G. . Klyve . D. W. . Near Wieferich primes up to 6.7 × 10¹⁵ . 2010 .
6. http://www.primegrid.com/forum_thread.php?id=3008&nowrap=true#45946 Wall–Sun–Sun Prime Search project
7. http://www.primegrid.com/forum_thread.php?id=7436&nowrap=true#107809
8. https://www.primegrid.com/forum_forum.php?id=127 Message boards : Wieferich and Wall-Sun-Sun Prime Search
9. https://www.primegrid.com/server_status_subprojects.php Subproject status
10. Crandall. R. . Dilcher . k. . Pomerance. C. . 447 . A search for Wieferich and Wilson primes . Mathematics of Computation . 66 . 1997 . 217 . 1997MaCom..66..433C .
11. Klaška . Jiří . A search for Tribonacci–Wieferich primes . Acta Mathematica Universitatis Ostraviensis . 16 . 1 . 15–20 . 2008 .
12. Reginald McLean and PrimeGrid, WW Statistics
13. S. Falcon, A. Plaza . k-Fibonacci sequence modulo m . Chaos, Solitons & Fractals . 41 . 1 . 2009 . 497–504 . 10.1016/j.chaos.2008.02.014. 2009CSF....41..497F .