Fibonacci Prime Test

A Fibonacci prime is the rarest special number in elementary arithmetic: it must be both a member of the Fibonacci sequence and a prime. The list starts 2, 3, 5, 13, 89, 233, 1597, 28657 — sparse values at sparse indexes, and because 8 and 21 already show composite Fibonacci numbers, the intersection is genuinely narrow. Even deciding whether a number like 433494437 qualifies needs two tests, and an eyeball does neither well.

Fibonacci Prime Test applies both checks exactly, in your browser. Membership uses the Lucas identity (5n squared ± 4 is a square) with BigInt square roots, and primality uses a deterministic-for-all-practical-sizes Miller-Rabin test. Every input gets a full classification — Fibonacci prime, Fibonacci composite, or not Fibonacci — with its index, and batches of 10,000 values run in one pass. Nothing is uploaded, ever.

• Two independent exact tests: Lucas square-test for membership, Miller-Rabin (twelve bases) for primality.
• Five-way classification: Fibonacci prime, Fibonacci composite, prime but not Fibonacci, not Fibonacci, and the special-cased 0 and 1.
• Index display: Confirmed members show k with F(k) = n — and the odd-index pattern behind Fibonacci primes.
• BigInt everywhere: Hundreds of digits classified exactly, with zero floating-point involvement.
• Batch testing: Up to 10,000 values per run, each verdict independent and detailed.
• Copy or download: Take the classified list to the clipboard or save it as a text file.
• Reference tab: The first 26 known Fibonacci primes, with the even-index factorization note.
• Free and private: Everything runs locally in the browser; nothing is uploaded or stored.

1. Enter numbers — one per line, or a batch separated by commas or spaces.
2. Read the five-way verdict — each value is classified against both tests.
3. Inspect the detail row — index for members, square-test note, and nearest neighbors.
4. Compare edge cases — try 8 (composite fib), 7 (prime, not fib), 89 (true Fibonacci prime).
5. Copy or download the results for reports, tests or teaching.

Example 1 — The twin primes of the sequence. Enter 2 and 3. Both return "Fibonacci prime": F(3) = 2 and F(4) = 3, the first two members of the list.

Example 2 — A composite member. Enter 21. The verdict is "Fibonacci, composite" — F(8), but 21 = 3 x 7, so membership alone is not enough.

Example 3 — The other side of the coin. Enter 7. The verdict is "not Fibonacci" — 7 is prime but never appears in the sequence, showing why both tests are needed.

Example 4 — The famous jump. Enter 514229. The verdict is "Fibonacci prime" at index 29 — one of the most recognizable large entries in the list.

Example 5 — A batch audit. Enter 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. The tool splits them into Fibonacci primes (5, 13, 89, 233) and composite members (8, 21, 34, 55, 144, 377) automatically.

• Both tests, exact: Membership and primality are each proven, never guessed.
• Full classification: Five distinct verdicts cover every possible input including the awkward 0 and 1.
• Practical certainty: Deterministic Miller-Rabin bases cover the entire range ordinary numbers occupy.
• Educational clarity: Indexes and factorization notes explain why each verdict is correct.
• Batch speed: Ten thousand candidates classified per run with per-line detail.
• Private and free: Everything runs in the browser; nothing is uploaded, stored or logged.