Wall-Sun-Sun prime

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English

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Alternative forms

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Etymology

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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

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

  1. (number theory) A (hypothetical) prime number such that divides , where is the Fibonacci sequence and is the th Pisano period (the period length of the Fibonacci sequence reduced modulo ).
    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 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 satisfying a Wall–Sun–Sun prime. There is no known example of a Wall–Sun–Sun prime and the congruence , can only be checked through explicit powering computations.

Usage notes

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  • Definition (slightly expanded):
    • Consider the Fibonacci sequence . For any prime number , reducing the sequence modulo produces a periodic sequence. The period length of the reduced sequence is called the th Pisano period, denoted . Since , it follows that .
    • A Wall-Sun-Sun prime is a prime number such that .
  • Alternative definitions:
    • Denote by the rank of apparition modulo (the smallest such that ). For prime , it is known that , where is the Legendre symbol. Then:
      • A prime is a Wall-Sun-Sun prime if and only if .
      • A prime is a Wall-Sun-Sun prime if and only if .
      • A prime is a Wall-Sun-Sun prime if and only if .
      • A prime is a Wall-Sun-Sun prime if and only if , where is the th Lucas number.

Translations

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See also

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Further reading

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