Wall–Sun–Sun prime

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 be a prime number. When each term in the sequence of Fibonacci numbers is reduced modulo (i.e. subtracted by the largest multiple of less than it), the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted . Since , it follows that p divides . A prime p such that p2 divides is called a Wall–Sun–Sun prime.

+ Non-Wall–Sun–Sun primes Periodic sequence Quotient of 2 0, 1, 1 3 2 0.5 3 0, 1, 1, 2, 0, 2, 2, 1 8 21 2. 5 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1 20 6,765 270.6 7 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1 16 987 20. 11 0, 1, 1, 2, 3, 5, 8, 2, 10, 1 10 55 0.

A prime is a Wall–Sun–Sun prime if and only if the quotient of divided by p2 is a natural number.

Equivalent definitions

If denotes the rank of apparition modulo (i.e., is the smallest positive index such that divides ), then a Wall–Sun–Sun prime can be equivalently defined as a prime such that divides .

For a prime p ≠ 2, 5, the rank of apparition is known to divide , where the Legendre symbol has the values

This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes such that divides the Fibonacci number .

A prime is a Wall–Sun–Sun prime if and only if .

A prime is a Wall–Sun–Sun prime if and only if , where is the -th Lucas number.

McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes. In particular, let ; then the following are equivalent:

Existence

In a study of the Pisano period , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than . In 1960, he wrote:

It has since been conjectured that there are infinitely many Wall–Sun–Sun primes..

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

In December 2011, another search was started by the PrimeGrid project;Wall–Sun–Sun Prime Search project at PrimeGrid however, it was suspended in May 2017. at PrimeGrid In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously.Message boards : Wieferich and Wall-Sun-Sun Prime Search at PrimeGrid The project ended in December 2022, proving that any Wall–Sun–Sun prime must exceed (about ).Subproject status at PrimeGrid

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 Wiles's 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(m) is the least positive integer k satisfying [Tk,Tk+1,Tk+2] ≡ [T0, T1, T2] (mod m) and Tn denotes the n-th tribonacci number. No tribonacci–Wieferich prime exists below 1011.

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

Near-Wall–Sun–Sun primes A prime p such that 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.Reginald McLean and PrimeGrid, WW Statistics A dozen cases are known where A = ±1 .

Wall–Sun–Sun primes with discriminant D Wall–Sun–Sun primes can be considered for 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. In this definition, the prime p should be odd and not divide D.

It is conjectured that for every fundamental discriminant D not equal to 1, 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 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 , where Fk(n) = Un(k, −1) is a Lucas sequence of the first kind with discriminant D = k2 + 4 and 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 , where is the Legendre 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 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

References

Wall–Sun–Sun prime

Bron en licentie