Definitions

• The generalized $a,b$-Fibonacci sequence (the $n$th term of which is denoted ${^a _b}F_n$) is defined recursively by the following statements:
  • ${^a _b}F_0 = 0$.
  • ${^a _b}F_1 = 1$.
  • ${^a _b}F_n = a*{^a _b}F_{n-1} + b*{^a _b}F_{n-2}$, $n>2$.
  Note that the regular Fibonacci numbers can be denoted by ${^1 _1}F_n$ in this notaion.
• ${^a _b}z(m) = c$ means that $c$ has the following properties:
  • $c>0$.
  • ${^a _b}F_c\equiv 0\pmod{m}$.
  • $c$ is the least positive integer satisfying these properties.
• ${^a _b}t(m) = d$ means that $d$ has the following properties:
  • $d>0$.
  • ${^a _b}F_d\equiv 0\pmod{m}$.
  • ${^a _b}F_{d+1}\equiv 1\pmod{m}$.
  • $d$ is the least positive integer satisfying these properties.
  Note: ${^a _b}t(m)$ is then the function that gives the period of the generalized $a,b$-Fibonacci sequence in modulo $m$.