What is a Fibonacci prime?

A Fibonacci number that also happens to be a prime number. The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..., and among these, 2, 3, 5, 13, and 89 are Fibonacci primes because each is both a Fibonacci term and a prime number.

Why is there a lower count cap than the other Fibonacci tools?

Fibonacci primes get rare fast, only a couple dozen are known below index 3000, and confirming primality on a hundreds-of-digits-long Fibonacci term takes real work. The count is capped at 20 so the search always completes within a bounded, fast search range.

How is primality tested on such large numbers?

With a Miller-Rabin primality test, after ruling out small factors directly. Miller-Rabin is the standard practical approach for testing very large numbers, it's what production cryptography libraries use, and with the fixed set of witness bases this tool uses it is correct for every number in the tool's search range.

Except for 3, is there a pattern to which Fibonacci numbers are prime?

Yes, with the single exception of F(4) = 3, every known Fibonacci prime has a prime index in the Fibonacci sequence itself (5 is F(5), 13 is F(7), 89 is F(11), and so on). The reverse isn't guaranteed though, having a prime index doesn't mean the Fibonacci number itself is prime.

Fibonacci Prime Generator - Fibonacci Numbers That Are Also Prime

Overview

Introduction

Fibonacci primes sit at the intersection of two of number theory's most-studied topics, the Fibonacci sequence and prime numbers, and they get progressively rarer the further out you look.

This tool searches forward through the Fibonacci sequence, testing each term for primality, and returns the first N that qualify.

What Is Generate Fibonacci Primes?

A generator that filters the Fibonacci sequence down to only the terms that are also prime numbers: 2, 3, 5, 13, 89, 233, 1597, and so on.

It combines Fibonacci term generation with a Miller-Rabin primality test capable of handling the hundreds-of-digits-long numbers that appear deeper in the sequence.

How Generate Fibonacci Primes Works

The tool walks the Fibonacci sequence term by term using BigInt arithmetic, testing each term for primality as it's produced.

Primality testing first rules out small prime factors directly, then applies a Miller-Rabin test with a fixed set of witness bases for larger candidates; qualifying terms are collected until N have been found or the search range is exhausted.

When To Use Generate Fibonacci Primes

Use it whenever you need the specific subsequence of Fibonacci numbers that are prime, for number theory coursework, puzzle-solving, or curiosity about how the two properties overlap.

If you just need plain Fibonacci numbers without the primality filter, use Fibonacci Number Generator instead.

Often used alongside Generate Fibonacci Numbers, Generate Prime Numbers and Generate Lucas Primes.

Features

Advantages

Handles the large numbers involved correctly, Fibonacci terms in the hundreds of digits are tested exactly as reliably as small ones.
Saves the work of generating a Fibonacci sequence yourself and manually checking each term for primality.
Searches automatically as far as needed within its range, you just specify how many results you want.

Limitations

Capped at 20 results, since known Fibonacci primes become sparse and the search has to stay within a range that resolves in a reasonable time.
Uses Miller-Rabin, a probabilistic primality test, though the fixed witness set used here is exact for every number size this tool can produce.

Examples

First 6 Fibonacci primes

Input

Output

2, 3, 5, 13, 89, 233

These are the Fibonacci numbers at indices 3, 4, 5, 7, 11, and 13 that also happen to be prime.

First 3 Fibonacci primes

Input

Output

2, 3, 5

Best Practices & Notes

Best Practices

Keep expectations realistic about count size, Fibonacci primes thin out quickly, so this isn't a tool for generating hundreds of them.
If the search reports it couldn't find enough terms, try a smaller count rather than assuming something's broken, it means the next Fibonacci prime lies beyond the tool's search range.

Developer Notes

The Miller-Rabin implementation uses the fixed witness set {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}, which is deterministic for every integer below roughly 3.3 x 10^24 and an extremely reliable probabilistic test well beyond that range; combined with a hard search cap on how many Fibonacci indices are scanned, this keeps worst-case runtime bounded no matter what count is requested.

Generate Fibonacci Primes Use Cases

Studying which Fibonacci numbers are also prime for number theory coursework
Generating known Fibonacci-prime reference values for testing a primality-testing implementation
Exploring the (still unproven) conjecture that infinitely many Fibonacci primes exist

Common Mistakes

Requesting a count well above 20 expecting it to just take longer, the tool caps the count outright rather than let a search run indefinitely.
Assuming every Fibonacci number with a prime index is itself prime, that's not guaranteed, it's only true in the reverse direction for the known Fibonacci primes above 3.

Tips

Compare this list against Lucas Prime Generator to see how the two closely related sequences differ in which terms are prime.
Cross-check a specific value against Prime Number Sequence Generator's output if you want to confirm a term is prime independent of its Fibonacci origin.

Generate Fibonacci Primes