Wall-Sun-Sun prime
(Redirected from Wall–Sun–Sun prime)
English edit
Alternative forms edit
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 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 edit
- 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.
- Denote by the rank of apparition modulo (the smallest such that ). For prime , it is known that , where is the Legendre symbol. Then:
Translations edit
conjectured prime number of a certain type
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See also edit
Further reading edit
- Fibonacci prime § Wall–Sun–Sun primes on Wikipedia.Wikipedia