Definitions

• The Fibonacci-$q$ sequence (the $n$th term of which is denoted $F_{n,q}$) is defined recursively by the following statements:
  • $F_{0,q} = 0$.
  • $F_{1,q} = q$.
  • $F_{n,q} = F_{n-1,q} + F_{n-2,q}$, $n>2$.
  Note that the regular Fibonacci numbers can be denoted by $F_{n,1}$ in this notaion.
• $z_q (a) = c$ means that $c$ has the following properties:
  • $c>0$.
  • $F_{c,q}\equiv 0\pmod{a}$.
  • $c$ is the least positive integer satisfying these properties.
• $t_q (a) = d$ means that $d$ has the following properties:
  • $d>0$.
  • $F_{d,q}\equiv 0\pmod{a}$.
  • $F_{d+1,q}\equiv q\pmod{a}$.
  • $d$ is the least positive integer satisfying these properties.
  Note: $t_q (a)$ is then the function that gives the period of the Fibonacci-$q$ sequence in modulo $a$.
• $numz(a) = g$ means that $g$ has the following properties:
  • $g$ is the number of $F_n$ that are congruent to $0\pmod{a}$, where $n$ goes from $1$ to $t(a)$, inclusive.
• $numz_q (a) = h$ means that $h$ has the following properties:
  • $h$ is the number of $F_{n,q}$ that are congruent to $0\pmod{a}$, where $n$ goes from $1$ to $t_q (a)$, inclusive.
• $\phi(n) = j$ (known as the Euler phi function) means that $j$ has the following properties:
  • $j$ is the number of $k$ such that $(k,n)=1$, where $k$ goes from $1$ to $n-1$, inclusive.