# Composite Fibonacci Numbers with Prime Index

## Theorem

The sequence of composite Fibonacci numbers with a prime index begins:

$4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$

The corresponding sequence of prime indices begins:

$19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$

## Proof

By observation:

 $\displaystyle F_{19}$ $=$ $\displaystyle 4181$ $\displaystyle$ $=$ $\displaystyle 37 \times 113$

 $\displaystyle F_{31}$ $=$ $\displaystyle 1 \, 346 \, 269$ $\displaystyle$ $=$ $\displaystyle 557 \times 2417$