rank of apparition

rank of apparition (plural ranks of apparition)

1. (number theory) Given a positive integer m and a divisibility sequence S_k, the smallest index k such that S_k is divisible by m;
   such an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).

   Synonym: (for the Fibonacci sequence) Fibonacci entry point

   • 1986, Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, page 114:

     Let ${\displaystyle (W_{n})}$ be an EDS^{[Elliptic Divisibility Sequence]} associated to an elliptic curve ${\displaystyle E/K}$ and a nonzero point ${\displaystyle P\in E(K)}$ of finite order. Let ${\displaystyle r\geq 2}$ be the smallest index such that ${\displaystyle W_{r}=0}$. (The number ${\displaystyle r}$ is called the rank of apparition of the sequence.)
     […]
     Suppose that ${\displaystyle K}$ is a finite field and that the rank of apparition ${\displaystyle r}$ of ${\displaystyle (W_{n})}$ is at least 4.

   • 1996, D. L. Wells, Residue Counts Modulo Three for the Fibonacci Triangle, G. E. Bergum, Andreas N. Philippou, Alwyn F. Horadam (editors), Applications of Fibonacci Numbers, Kluwer Academic, Softcover reprint, page 535,

     A similar identification between Pascal's Triangle modulo p and the Fibonacci Triangle modulo p can be made for primes p which have the length of the period equal to twice their rank of apparition in the Fibonacci Sequence.

   • 2013, E. L. Roettger, H. C. Williams, R. K. Guy, “Some Extensions of the Lucas Functions”, in Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin, editors, Number Theory and Related Fields, Springer, page 303:

     Indeed, as shown in [18, Theorem 4.27], there exist sequences ${\displaystyle \{U_{n}\}}$ and primes ${\displaystyle p}$ for which ${\displaystyle p}$ has three ranks of apparition. In the previous section, we showed that if ${\displaystyle p=2}$, then ${\displaystyle p}$ has no more than two ranks of apparition in ${\displaystyle \{U_{n}\}}$.

• Typically, the integer whose rank of apparition is wanted (here called m) is specified to be a prime number (and called p). This reflects the fact that the principal issue at hand is divisibility.

• Fibonacci prime § Rank of apparition on Wikipedia.Wikipedia
• Wall–Sun–Sun prime on Wikipedia.Wikipedia