When each term in the sequence of Fibonacci numbers
, the result is a periodic sequence.
The (minimal) period length of this sequence is called the Pisano period and denoted
denotes the rank of apparition modulo
For a prime p ≠ 2, 5, the rank of apparition
has the values This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes
[2]: 42 McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes.
; 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:[4] The most perplexing problem we have met in this study concerns the hypothesis
We have run a test on digital computer which shows that
The question is closely related to another one, "can a number
", for which rare cases give an affirmative answer (e.g.,
); hence, one might conjecture that equality may hold for some exceptional
[5] 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.
[6] In December 2011, another search was started by the PrimeGrid project;[7] however, it was suspended in May 2017.
[8] In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously.
[9] The project ended in December 2022, definitely proving that any Wall–Sun–Sun prime must exceed
[10] Wall–Sun–Sun primes are named after Donald Dines Wall,[4][11] 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.
[12] 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.
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.
[13] 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).
with small |A| is called near-Wall–Sun–Sun prime.
[14] A dozen cases are known where A = ±1 (sequence A347565 in the OEIS).
Wall–Sun–Sun primes can be considered for the field
[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
, 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.[15] For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following.