Equivalent definitions

If ${\displaystyle \alpha (m)}$ denotes the rank of apparition modulo ${\displaystyle m}$ (i.e., ${\displaystyle \alpha (m)}$ is the smallest positive index such that ${\displaystyle m}$ divides ${\displaystyle F_{\alpha (m)}}$), then a Wall–Sun–Sun prime can be equivalently defined as a prime ${\displaystyle p}$ such that ${\displaystyle p^{2}}$ divides ${\displaystyle F_{\alpha (p)}}$.

For a prime p ≠ 2, 5, the rank of apparition ${\displaystyle \alpha (p)}$ is known to divide ${\displaystyle p-\left({\tfrac {p}{5}}\right)}$, where the Legendre symbol ${\displaystyle \textstyle \left({\frac {p}{5}}\right)}$ has the values

${\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}1&{\text{if }}p\equiv \pm 1{\pmod {5}};\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}$

This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes ${\displaystyle p}$ such that ${\displaystyle p^{2}}$ divides the Fibonacci number ${\displaystyle F_{p-\left({\frac {p}{5}}\right)}}$.^[1]

A prime ${\displaystyle p}$ is a Wall–Sun–Sun prime if and only if ${\displaystyle \pi (p^{2})=\pi (p)}$.

A prime ${\displaystyle p}$ is a Wall–Sun–Sun prime if and only if ${\displaystyle L_{p}\equiv 1{\pmod {p^{2}}}}$, where ${\displaystyle L_{p}}$ is the ${\displaystyle p}$-th Lucas number.^[2]: 42

McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes.^[3] In particular, let ${\displaystyle \epsilon =\left({\tfrac {p}{5}}\right)}$; then the following are equivalent:

• ${\displaystyle F_{p-\epsilon }\equiv 0{\pmod {p^{2}}}}$
• ${\displaystyle L_{p-\epsilon }\equiv 2\epsilon {\pmod {p^{4}}}}$
• ${\displaystyle L_{p-\epsilon }\equiv 2\epsilon {\pmod {p^{3}}}}$
• ${\displaystyle F_{p}\equiv \epsilon {\pmod {p^{2}}}}$
• ${\displaystyle L_{p}\equiv 1{\pmod {p^{2}}}}$

Near-Wall–Sun–Sun primes

A prime p such that ${\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv Ap{\pmod {p^{2}}}}$ with small |A| is called near-Wall–Sun–Sun prime.^[3] Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes. PrimeGrid recorded cases with |A| ≤ 1000.^[14] A dozen cases are known where A = ±1 (sequence A347565 in the OEIS).

Wall–Sun–Sun primes with discriminant D

Wall–Sun–Sun primes can be considered for the field ${\displaystyle Q_{\sqrt {D}}}$ 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.^[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 ${\displaystyle (P,Q)=(k,-1)}$ 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 ${\displaystyle F_{k}(\pi _{k}(p))}$, where F_k(n) = U_n(k, −1) is a Lucas sequence of the first kind with discriminant D = k² + 4 and ${\displaystyle \pi _{k}(p)}$ 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.

• p² divides ${\displaystyle F_{k}\left(p-\left({\tfrac {D}{p}}\right)\right)}$, where ${\displaystyle \left({\tfrac {D}{p}}\right)}$ is the Kronecker symbol;
• V_p(k, −1) ≡ k (mod p²), where V_n(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, ... (sequence A271782 in the OEIS)