Wall-Sun-Sun prime

• Wall-Sun-Sun prime number

Named after American mathematician Donald Dines Wall and Chinese mathematicians Sun Zhihong and Sun Zhiwei, who have all contributed to the study of such primes.

Wall-Sun-Sun prime (plural Wall-Sun-Sun primes)

1. (number theory) A (hypothetical) prime number ${\displaystyle p}$ such that ${\displaystyle p^{2}}$ divides ${\displaystyle F_{\pi (p)}}$, where ${\displaystyle F_{n}}$ is the Fibonacci sequence and ${\displaystyle \pi (p)}$ is the ${\displaystyle p}$th Pisano period (the period length of the Fibonacci sequence reduced modulo ${\displaystyle p}$).

   Synonym: Fibonacci-Wieferich prime

   Wall-Sun-Sun primes are conjectured to exist, but no example has yet been found.

   • 1996, Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, page 113:

     Not a single Wall-Sun-Sun prime ${\displaystyle 5<p<1.6\times 10^{12}}$ exists [McIntosh 1995]. Carry out a search for Wall-Sun-Sun primes somewhere above this limit.

   • 2020, Dorin Andrica, Ovidiu Bagdasar, Recurrent Sequences, Springer, page 88:

     Crandall et al. called in [56] such a prime number ${\displaystyle p}$ satisfying ${\displaystyle p^{2}\vert F_{p-({\frac {p}{5}})}}$ a Wall–Sun–Sun prime. There is no known example of a Wall–Sun–Sun prime and the congruence ${\displaystyle F_{p-({\frac {p}{5}})}\equiv 0{\pmod {p^{2}}}}$, can only be checked through explicit powering computations.

• Definition (slightly expanded):
  • Consider the Fibonacci sequence ${\displaystyle F_{n}}$. For any prime number ${\displaystyle p}$, reducing the sequence modulo ${\displaystyle p}$ produces a periodic sequence. The period length of the reduced sequence is called the ${\displaystyle p}$th Pisano period, denoted ${\displaystyle \pi (p)}$. Since ${\displaystyle F_{0}=0}$, it follows that ${\displaystyle p\vert F_{\pi (p)}}$.
  • A Wall-Sun-Sun prime is a prime number ${\displaystyle p}$ such that ${\displaystyle p^{2}\vert F_{\pi (p)}}$.
• Alternative definitions:
  • Denote by ${\displaystyle \alpha (m)}$ the rank of apparition modulo ${\displaystyle m}$ (the smallest ${\displaystyle k}$ such that ${\displaystyle m\vert F_{k}}$). For prime ${\displaystyle p\neq 2,5}$, it is known that ${\displaystyle \alpha (p)\vert p-\left({\tfrac {p}{5}}\right)}$, where ${\displaystyle \textstyle \left({\frac {p}{5}}\right)}$ is the Legendre symbol. Then:
    • A prime ${\displaystyle p}$ is a Wall-Sun-Sun prime if and only if ${\displaystyle p^{2}\vert F_{\alpha (p)}}$.
    • A prime ${\displaystyle p}$ is a Wall-Sun-Sun prime if and only if ${\displaystyle p^{2}\vert F_{p-\left({\frac {p}{5}}\right)}}$.
    • 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.

• Pisano period
• rank of apparition

• Fibonacci prime § Wall–Sun–Sun primes on Wikipedia.Wikipedia