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Wall–Sun–Sun prime

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Wall–Sun–Sun prime

Wall–Sun–Sun prime
Named after Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun
Publication year 1992
Number of known terms 0
Conjectured number of terms Infinite
Open problem in mathematics:
Are there any Wall–Sun–Sun primes? If yes, is there an infinite number of them?
(more open problems in mathematics)

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 1
  • Existence 2
  • History 3
  • Generalizations 4
    • Near-Wall-Sun-Sun primes 4.1
    • Wall-Sun-Sun primes with discriminant D 4.2
  • See also 5
  • References 6
  • Further reading 7
  • External links 8

Definition

A prime p ≠ 2, 5 is called a Wall–Sun–Sun prime if p2 divides the Fibonacci number F_{p - \left(\frac\right)}, where the Legendre symbol \textstyle\left(\frac\right) has the values

\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}

Equivalently, a prime p is a Wall–Sun–Sun prime iff Lp ≡ 1 (mod p2), where Lp is the p-th Lucas number.[1]:42

A k-Wall-Sun-Sun prime is defined as a prime p such that p2 divides the k-Fibonacci number (a Lucas sequence Un with (P, Q) = (k, -1)) F_k({p - \left(\frac\right)}), where \left(\frac\right) is the Legendre symbol. For example, 241 is a k-Wall-Sun-Sun prime for k = 3. Thus, a prime p is a k-Wall-Sun-Sun prime iff Vk(p) ≡ 1 (mod p2), where Vn is a Lucas sequence with (P, Q) = (k, -1).

Least n-Wall-Sun-Sun prime are

13, 241, 2, 3, 191, 5, 2, 3, 2683, ... (start with n = 2)

Existence

It has been conjectured that there are infinitely many Wall–Sun–Sun primes.[2] No Wall–Sun–Sun primes are known as of October 2014.

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×1014.[3] Dorais and Klyve extended this range to 9.7×1014 without finding such a prime.[4] In December 2011, another search was started by the PrimeGrid project.[5] As of October 2014, PrimeGrid has extended the search limit to 2.8×1016 and continues.[6]

History

Wall–Sun–Sun primes are named after Donald Dines Wall,[7] 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.[8] 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 [Th,Th+1,Th+2] ≡ [T0, T1, T2] (mod m) and Tn denotes the n-th Tribonacci number. No Tribonacci-Wieferich prime exists below 1011.[9]

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

Near-Wall-Sun-Sun primes

A prime p such that F_{p - \left(\frac\right)}=Ap\ (mod\ p^2) with small |A| is called near-Wall–Sun–Sun prime.[10] Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes.

Wall-Sun-Sun primes with discriminant D

It is in the field Q \sqrt{D} with discriminant D. In the original Wall–Sun–Sun prime case, D = 5, more generalized, a Lucas-Wieferich prime to (P, Q) is a Wieferich prime to base Q and a Wall–Sun–Sun prime with discriminant D = P2 + 4Q.[11]

In this definition, a generalized Wall–Sun–Sun prime p should be odd and not divide D. It's a conjecture that for every natural number D, there are infinitely many Wall–Sun–Sun primes with discriminant D.

D Wall–Sun–Sun primes with discriminant D 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 7, 523, ...
7 ...
8 13, 31, 1546463, ...
9 5, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes except 3)
10 191, 643, 134339, 25233137, ...
11 ...
12 103, 2297860813, ...
13 241, ...
14 6707879, 93140353, ...
15 181, 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, 7, 11, 13, 17, 19, 23, 29, ... (All odd primes except 5)
26 2683, 3967, 18587, ...
27 103, 2297860813, ...
28 ...
29 3, 11, ...
30 ...

See also

References

  1. ^ Andrejić, V. (2006). "On Fibonacci powers" (PDF). Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44.  
  2. ^ Klaška, Jiří (2007), "Short remark on Fibonacci−Wieferich primes", Acta Mathematica Universitatis Ostraviensis 15 (1): 21–25 .
  3. ^ McIntosh, R. J.; Roettger, E. L. (2007). "A search for Fibonacci−Wieferich and Wolstenholme primes" (PDF).  
  4. ^ Dorais, F. G.; Klyve, D. W. (2010). "15"Near Wieferich primes up to 6.7 × 10 (PDF). 
  5. ^ Wall–Sun–Sun Prime Search project at PrimeGrid
  6. ^ Wall-Sun-Sun Prime Search statistics at PrimeGrid
  7. ^ Wall, D. D. (1960), "Fibonacci Series Modulo m",  
  8. ^ Sun, Zhi-Hong; Sun, Zhi-Wei (1992), "Fibonacci numbers and Fermat’s last theorem" (PDF),  
  9. ^ Klaška, Jiří (2008). "A search for Tribonacci-Wieferich primes". Acta Mathematica Universitatis Ostraviensis 16 (1): 15–20. 
  10. ^ McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes", Mathematics of Computation ( 
  11. ^ 2pThe Fibonacci sequence modulo

Further reading

  • Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29.  
  • Saha, Arpan; Karthik, C. S. (2011). "A Few Equivalences of Wall-Sun-Sun Prime Conjecture". Working paper.  

External links

  • 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)
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