Wall–Sun–Sun prime

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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[edit]

The period of Fibonacci numbers modulo prime p is called the Pisano period and denoted . It follows that p divides . A prime p such that p2 divides is called a Wall–Sun–Sun prime.

For a prime p ≠ 2, 5, the Pisano period is known to divide , where the Legendre symbol has the values

This observation gives rise to an equivalent definition that a prime p is a Wall–Sun–Sun prime if p2 divides the Fibonacci number .[1]

Equivalently, a prime p is a Wall–Sun–Sun prime if Lp ≡ 1 (mod p2), where Lp is the p-th Lucas number.[2]:42

Existence[edit]

Question dropshade.png 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 , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than . He writes:[3]

The most perplexing problem we have met in this study concerns the hypothesis . We have run a test on digital computer which shows that for all up to ; however, we cannot prove that is impossible. The question is closely related to another one, "can a number have the same order mod and mod ?", for which rare cases give an affirmative answer (e.g., ; ); hence, one might conjecture that equality may hold for some exceptional .

It has since been conjectured that there are infinitely many Wall–Sun–Sun primes.[4] No Wall–Sun–Sun primes are known as of April 2016.

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×1014.[5] Dorais and Klyve extended this range to 9.7×1014 without finding such a prime.[6]

In December 2011, another search was started by the PrimeGrid project.[7] As of April 2016, PrimeGrid has extended the search limit to 1.9×1017 and continues.[8]

History[edit]

Wall–Sun–Sun primes are named after Donald Dines Wall,[3][9] 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.[10] 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.[11]

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

Near-Wall–Sun–Sun primes[edit]

A prime p such that with small |A| is called near-Wall–Sun–Sun prime.[12] Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes.

Wall–Sun–Sun primes with discriminant D[edit]

Wall–Sun–Sun primes can be considered in the field 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 corresponds to the k-Wall–Sun–Sun primes, for which Wall–Sun–Sun primes represent a special case with k = 1. The k-Wall–Sun–Sun primes can be explicitly defined as primes p such that p2 divides the k-Fibonacci number , where Fk(n) = Un(k, −1) is a Lucas sequence of first kind with discriminant D = k2 + 4 and 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 any of the following two:

  • p2 divides , where is the Kronecker symbol;
  • Vp(k, −1) ≡ k (mod p2), where Vn(k, −1) is a Lucas sequence of the second kind.

The smallest k-Wall–Sun–Sun prime 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 (OEISA013946) k-Wall–Sun–Sun primes notes
1 5 ...
2 2 13, 31, 1546463, ...
3 13 241, ...
4 5 2, 3, ... Since this is the second time for which D=5, thus plus the prime factors of 2*2−1 which does not divide 5. Since k is divisible by 4, thus plus the prime 2.
5 29 3, 11, ...
6 10 191, 643, 134339, 25233137, ...
7 53 5, ...
8 17 2, ... Since k is divisible by 4, thus plus the prime 2.
9 85 3, 204520559, ...
10 26 2683, 3967, 18587, ...
11 5 ... Since this is the third time for which D=5, thus plus the prime factors of 2*3−1 which does not divide 5.
12 37 2, 7, 89, 257, 631, ... Since k is divisible by 4, thus plus the prime 2.
13 173 3, 227, 392893, ...
14 2 3, 13, 31, 1546463, ... Since this is the second time for which D=2, thus plus the prime factors of 2*2−1 which does not divide 2.
15 229 29, 4253, ...
16 65 2, 1327, 8831, 569831, ... Since k is divisible by 4, thus plus the prime 2.
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, thus plus the prime 2.
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, thus plus the prime 2.
25 629 5, 7, 2687, ...
26 170 79, ...
27 733 3, 1663, ...
28 197 2, 1431615389, ... Since k is divisible by 4, thus plus the prime 2.
29 5 7, ... Since this is the fourth time for which D=5, thus plus the prime factors of 2*4−1 which does 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 3, ...
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[edit]

References[edit]

  1. ^ a b A.-S. Elsenhans, J. Jahnel (2010). "The Fibonacci sequence modulo p2 -- An investigation by computer for p < 1014". arXiv:1006.0824Freely accessible. 
  2. ^ Andrejić, V. (2006). "On Fibonacci powers" (PDF). Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A. 
  3. ^ a b Wall, D. D. (1960), "Fibonacci Series Modulo m", American Mathematical Monthly, 67 (6): 525–532, doi:10.2307/2309169 
  4. ^ Klaška, Jiří (2007), "Short remark on Fibonacci−Wieferich primes", Acta Mathematica Universitatis Ostraviensis, 15 (1): 21–25 .
  5. ^ McIntosh, R. J.; Roettger, E. L. (2007). "A search for Fibonacci−Wieferich and Wolstenholme primes" (PDF). Mathematics of Computation. 76 (260): 2087–2094. doi:10.1090/S0025-5718-07-01955-2. 
  6. ^ Dorais, F. G.; Klyve, D. W. (2010). "Near Wieferich primes up to 6.7 × 1015" (PDF). 
  7. ^ Wall–Sun–Sun Prime Search project at PrimeGrid
  8. ^ Wall–Sun–Sun Prime Search statistics at PrimeGrid
  9. ^ Crandall, R.; Dilcher, k.; Pomerance, C. (1997). "A search for Wieferich and Wilson primes". 66: 447. 
  10. ^ Sun, Zhi-Hong; Sun, Zhi-Wei (1992), "Fibonacci numbers and Fermat's last theorem" (PDF), Acta Arithmetica, 60 (4): 371–388 
  11. ^ Klaška, Jiří (2008). "A search for Tribonacci–Wieferich primes". Acta Mathematica Universitatis Ostraviensis. 16 (1): 15–20. 
  12. ^ McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes", Mathematics of Computation, AMS, 76 (260): 2087–2094, doi:10.1090/S0025-5718-07-01955-2, archived from the original (PDF) on 2010-12-10 
  13. ^ S. Falcon, A. Plaza (2009). "k-Fibonacci sequence modulo m". Chaos, Solitons & Fractals. 41 (1): 497–504. doi:10.1016/j.chaos.2008.02.014. 

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". arXiv:1102.1636Freely accessible. 

External links[edit]