Wall-Sun-Sun prime

English[edit]

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Wall–Sun–Sun prime

Alternative forms[edit]

Wall-Sun-Sun prime number

Etymology[edit]

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.

Noun[edit]

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

(number theory) A (hypothetical) prime number $p$ $p$ such that $p^{2}$ $p^{2}$ divides $F_{\pi (p)}$ $F_{\pi (p)}$ , where $F_{n}$ $F_{n}$ is the Fibonacci sequence and $\pi (p)$ $\pi (p)$ is the $p$ $p$ th Pisano period (the period length of the Fibonacci sequence reduced modulo $p$ $p$ ).
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 $5<p<1.6\times 10^{12}$ 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 $p$ satisfying $p^{2}\vert F_{p-({\frac {p}{5}})}$ a Wall–Sun–Sun prime. There is no known example of a Wall–Sun–Sun prime and the congruence $F_{p-({\frac {p}{5}})}\equiv 0{\pmod {p^{2}}}$ , can only be checked through explicit powering computations.

Usage notes[edit]

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