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, the result is a periodic sequence.
The 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.

Equivalent definitions

If denotes the rank of apparition modulo , 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:
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. No Wall–Sun–Sun primes are known.
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, however it was suspended in May of 2017.

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 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 ≡ 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, or p2 divides Pp+1, when p congruent to 3 or 5, 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.

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 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 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 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 = Un 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.
The smallest k-Wall–Sun–Sun primes for k = 2, 3,... are
ksquare-free part of D k-Wall–Sun–Sun primesnotes
15...None are known.
2213, 31, 1546463,...
313241,...
452, 3,...Since this is the second value of k for which D=5, the k-Wall–Sun–Sun primes include the prime factors of 2*2−1 that do not divide 5. Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
5293, 11,...
610191, 643, 134339, 25233137,...
7535,...
8172,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
9853, 204520559,...
10262683, 3967, 18587,...
115...Since this is the third value of k for which D=5, the k-Wall–Sun–Sun primes include the prime factors of 2*3−1 that do not divide 5.
12372, 7, 89, 257, 631,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
131733, 227, 392893,...
1423, 13, 31, 1546463,...Since this is the second value of k for which D=2, the k-Wall–Sun–Sun primes include the prime factors of 2*2−1 that do not divide 2.
1522929, 4253,...
16652, 1327, 8831, 569831,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
172931192625911,...
18823, 5, 11, 769, 256531, 624451181,...
1936511, 233, 165083,...
201012, 7, 19301,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
2144523, 31, 193,...
221223, 281,...
235333, 103,...
241452, 7, 11, 17, 37, 41, 1319,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
256295, 7, 2687,...
2617079,...
277333, 1663,...
281972, 1431615389,...Since k is divisible by 4, 2 is a k-Wall–Sun–Sun prime.
2957,...Since this is the fourth value of k for which D=5, the k-Wall–Sun–Sun primes include the prime factors of 2*4−1 that do not divide 5.
3022623, 1277,...

DWall–Sun–Sun primes with discriminant D OEIS sequence
13, 5, 7, 11, 13, 17, 19, 23, 29,...
213, 31, 1546463,...
3103, 2297860813,...
43, 5, 7, 11, 13, 17, 19, 23, 29,...
5...
6, 7, 523,...
7...
813, 31, 1546463,...
9, 5, 7, 11, 13, 17, 19, 23, 29,...
10191, 643, 134339, 25233137,...
11...
12103, 2297860813,...
13241,...
146707879, 93140353,...
15, 181, 1039, 2917, 2401457,...
163, 5, 7, 11, 13, 17, 19, 23, 29,...
17...
1813, 31, 1546463,...
1979, 1271731, 13599893, 31352389,...
20...
2146179311,...
2243, 73, 409, 28477,...
237, 733,...
247, 523,...
253,, 7, 11, 13, 17, 19, 23, 29,...
262683, 3967, 18587,...
27103, 2297860813,...
28...
293, 11,...
30...