Wall-Sun-Sun prime

English

Alternative forms

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Etymology

Named after American mathematician Template:W and Chinese mathematicians Template:W and Template:W, who have all contributed to the study of such primes.

Noun

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

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Usage notes

• 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 Template:M, denoted $\pi (p)$. Since $F_0=0$, it follows that $p|F_{\pi (p)}$.
  • A Wall-Sun-Sun prime is a prime number $p$ such that $p^2|F_{\pi (p)}$.
• Alternative definitions:
  • Denote by $\alpha (m)$ the Template:M modulo $m$ (the smallest $k$ such that $m|F_k$). For prime $p\ne 2,5$, it is known that $\alpha (p)|p-\left(\frac{p}{5}\right)$, where $\left(\frac{p}{5}\right)$ is the Legendre symbol. Then:
    • A prime $p$ is a Wall-Sun-Sun prime if and only if $p^2|F_{\alpha (p)}$.
    • A prime $p$ is a Wall-Sun-Sun prime if and only if $p^2|F_{p-\left(\frac{p}{5}\right)}$.
    • 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 $L_p\equiv 1(\mathrm{mod}{p}^2)$, where $L_p$ is the $p$th Lucas number.

Translations

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See also

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Further reading

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