Meaning of rank of apparition | Babel Free

Definitions

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

Examples

“Let (W#95;n) be an EDS^([Elliptic Divisibility Sequence]) associated to an elliptic curve E#47;K and a nonzero point P#92;inE(K) of finite order. Let r#92;ge 2 be the smallest index such that W#95;r#61;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#95;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.”

“Indeed, as shown in [18, Theorem 4.27], there exist sequences #92;#123;U#95;n#92;#125; and primes p for which p has three ranks of apparition. In the previous section, we showed that if p#61;2, then p has no more than two ranks of apparition in #92;#123;U#95;n#92;#125;.”

CEFR level