Wall-Sun-Sun prime

English

Alternative forms

Wall-Sun-Sun prime number

Etymology

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

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

(number theory) A (hypothetical) prime number ${\\displaystyle p}$ $\text{[math]}$ such that ${\\displaystyle p^{2}}$ $\text{[math]}$ divides ${\\displaystyle F_{\\pi (p)}}$ $\text{[math]}$ , where ${\\displaystyle F_{n}}$ $\text{[math]}$ is the Fibonacci sequence and ${\\displaystyle \\pi (p)}$ $\text{[math]}$ is the ${\\displaystyle p}$ $\text{[math]}$ th Pisano period (the period length of the Fibonacci sequence reduced modulo ${\\displaystyle p}$ $\text{[math]}$ ).
Wall-Sun-Sun primes are conjectured to exist, but no example has yet been found.
Synonym: Fibonacci-Wieferich prime
- 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.

Usage notes

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 ${\\displaystyle \\alpha (m)}$ $\text{[math]}$ the rank of apparition modulo ${\\displaystyle m}$ $\text{[math]}$ (the smallest ${\\displaystyle k}$ $\text{[math]}$ such that ${\\displaystyle m\\vert F_{k}}$ $\text{[math]}$ ). For prime ${\\displaystyle p\eq 2,5}$ $\text{[math]}$ , it is known that ${\\displaystyle \\alpha (p)\\vert p-\\left({\\tfrac {p}{5}}\\right)}$ $\text{[math]}$ , where ${\\displaystyle \\textstyle \\left({\\frac {p}{5}}\\right)}$ $\text{[math]}$ 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.