Wall–Sun–Sun prime

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List of unsolved problems in mathematics
Are there any Wall–Sun–Sun primes? If yes, is there an infinite number of them?

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[edit]

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

\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 if Lp ≡ 1 (mod p2), where Lp is the p-th Lucas number.[1]:42

Existence[edit]

It has been conjectured that there are infinitely many Wall–Sun–Sun primes.[2] No Wall–Sun–Sun primes are known as of March 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 March 2014, PrimeGrid has extended the search limit to 2.8×1016 and continues.[6]

History[edit]

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[edit]

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]

See also[edit]

References[edit]

  1. ^ Andrejić, V. (2006). "On Fibonacci powers". Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A. 
  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". Mathematics of Computation 76 (260): 2087–2094. doi:10.1090/S0025-5718-07-01955-2. 
  4. ^ Dorais, F. G.; Klyve, D. W. (2010). Near Wieferich primes up to 6.7 × 1015. 
  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", American Mathematical Monthly 67 (6): 525–532, doi:10.2307/2309169 
  8. ^ Sun, Zhi-Hong; Sun, Zhi-Wei (1992), "Fibonacci numbers and Fermat’s last theorem", Acta Arithmetica 60 (4): 371–388 
  9. ^ Klaška, Jiří (2008). "A search for Tribonacci-Wieferich primes". Acta Mathematica Universitatis Ostraviensis 16 (1): 15–20. 

Further reading[edit]

  • Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29. ISBN 0-387-94777-9. 
  • Saha, Arpan; Karthik, C. S. (2011). "A Few Equivalences of Wall-Sun-Sun Prime Conjecture". Working paper. arXiv:1102.1636. 

External links[edit]