MathNet 2017-part-07 — 172 problems (EN–VI) Question 6

The Fibonacci sequence is defined as follows: $F_{0}=0$, $F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0 \pmod{127}$ and $F_{m+1} \equiv 1 \pmod{127}$.