rank of apparition

English

Noun

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 $(W_{n})$ be an EDS^{[Elliptic Divisibility Sequence]} associated to an elliptic curve $E/K$ and a nonzero point $P\in E(K)$ of finite order. Let $r\geq 2$ be the smallest index such that $W_{r}=0$. (The number $r$ is called the rank of apparition of the sequence.)
     […]
     Suppose that $K$ is a finite field and that the rank of apparition $r$ of $(W_{n})$ 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 $\{U_{n}\}$ and primes $p$ for which $p$ has three ranks of apparition. In the previous section, we showed that if $p=2$, then $p$ has no more than two ranks of apparition in $\{U_{n}\}$.

Usage notes

• 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.

Translations

smallest index k such that F_k is divisible by a specified number

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Further reading

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