Definitions

• The $k$-Fibonacci sequence (the $n$th term of which is denoted $F_n^k$) is defined recursively by the following statements:
  • If $2-k\leq n\leq 0, n\in{\bf Z}$, then $F_n^k=0$.
  • $F_1^k=1$.
  • $F_n^k=\sum_{i=1}^k F_{n-i}^k$, $n>0$.
• $z^k (a)=c$ means that $c$ has the following properties:
  • $c>0$.
  • For all $i$ such that $c-(k-2)\leq i\leq c, i\in{\bf Z}$, it follows that $F_i^k\equiv 0\pmod{a}$.
  • $c$ is the least positive integer satisfying these properties.
• $t^k (a)=d$ means that $d$ has the following properties:
  • $d>0$
  • For all $i$ such that $d-(k-2)\leq i\leq d, i\in{\bf Z}$, it follows that $F_i^k\equiv 0\pmod{a}$.
  • $F_{d+1}^k\equiv 1\pmod{a}$.
  • $d$ is the least positive integer satisfying these properties.
• $b^k (a)=e$ means that $e$ has the following properties:
  • $F_{z_k (a)-(k-1)}^k\equiv e\pmod{a}$.