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:

\(\displaystyle F_{19}\) \(=\) \(\displaystyle 4181\)
\(\displaystyle \) \(=\) \(\displaystyle 37 \times 113\)


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



Sources