Wall-Sun-Sun prime

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

Noun

 * 1)  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 $$p$$).

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, 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 $$\alpha(m)$$ the modulo $$m$$ (the smallest $$k$$ such that $$m \vert F_k$$). For prime $$p \ne 2, 5$$, it is known that $$\alpha(p) \vert p - \left(\tfrac{p}{5}\right)$$, where $$\textstyle\left(\frac{p}{5}\right)$$ 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.

Translations

 * French: nombre premier de Wall-Sun-Sun
 * German: Wall-Sun-Sun-Primzahl
 * Italian: primo di Wall-Sun-Sun