Definitions

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