Wall–Sun–Sun prime

Let p be a prime number. When each term in the sequence of Fibonacci numbers F_n is reduced modulo p (i.e. subtracted by the largest multiple of p less than it), the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p). Since F_0 = 0, it follows that p divides F_{\pi(p)}. A prime p such that p2 divides F_{\pi(p)} is called a Wall–Sun–Sun prime. Equivalent definitions If \alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index such that m divides F_{\alpha(m)}), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p^2 divides F_{\alpha(p)}. For a prime p ≠ 2, 5, the rank of apparition \alpha(p) is known to divide p - \left(\tfrac{p}{5}\right), where the Legendre symbol \textstyle\left(\frac{p}{5}\right) has the values :\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} This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes p such that p^2 divides the Fibonacci number F_{p - \left(\frac{p}{5}\right)}. A prime p is a Wall–Sun–Sun prime if and only if \pi(p^2) = \pi(p). A prime p is a Wall–Sun–Sun prime if and only if L_p \equiv 1 \pmod{p^2}, where L_p is the p-th Lucas number. McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes. In particular, let \epsilon = \left(\tfrac{p}{5}\right); then the following are equivalent: • F_{p - \epsilon} \equiv 0 \pmod{p^2} • L_{p - \epsilon} \equiv 2\epsilon \pmod{p^4} • L_{p - \epsilon} \equiv 2\epsilon \pmod{p^3} • F_p \equiv \epsilon \pmod{p^2} • L_p \equiv 1 \pmod{p^2} == Existence ==

In a study of the Pisano period k(p), Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than 10000. In 1960, he wrote: 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 2017. In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. The project ended in December 2022, proving that any Wall–Sun–Sun prime must exceed 2^{64} (about 18\cdot 10^{18}). == 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 F_{p - \left(\frac{p}{5}\right)} \equiv Ap \pmod{p^2} with small |A| is called near-Wall–Sun–Sun prime. 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 Q_{\sqrt{D}} 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. For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following. • p2 divides F_k\left(p - \left(\tfrac{D}{p}\right)\right), where \left(\tfrac{D}{p}\right) 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 ==