# Composite Fibonacci Numbers with Prime Index

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## Theorem

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

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

This sequence is A050937 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The corresponding sequence of prime indices begins:

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

This sequence is A038672 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

By observation:

\(\ds F_{19}\) | \(=\) | \(\ds 4181\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 37 \times 113\) |

\(\ds F_{31}\) | \(=\) | \(\ds 1 \, 346 \, 269\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 557 \times 2417\) |

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $4181$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4181$