Fibonacci Prime Numbers Generator

Most people can name composite Fibonacci numbers — 8, 21, 34 — but the sequence hides primes at such sparse indexes that they feel almost accidental: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229. The pattern of their appearance is governed by index arithmetic — even indexes beyond 4 automatically factor, so all candidates are odd — and the density is so low that mathematicians still do not know whether the list ever ends.

Fibonacci Prime Numbers Generator finds numbers that are both Fibonacci and prime, exactly and in order. Generates the first N (up to 25) or every prime with index under your chosen limit, with each row carrying the Fibonacci index and the digit count, and every value rendered as exact BigInt. A Miller-Rabin primality test runs in the browser, so nothing leaves your machine. Copy or download the set in one click.

• Exact values: Every prime is a BigInt Fibonacci value — no rounding, no scientific notation, no truncation.
• Two generation modes: First 25 by count, or every prime under an index limit you choose.
• Index shortcuts: Odd indexes only — even indexes above 4 factor by F(2m) = F(m) x L(m) and are skipped.
• Miller-Rabin in the browser: Twelve prime bases, deterministically exact below 3.3 x 10^24.
• Rich rows: Index, value and digit count for every prime, in ascending order.
• Reproducible: Identical output on identical runs — perfect for worksheets and cross-checks.
• Copy or download: Values-only list for spreadsheets, tests and further analysis.
• Reference tab: The known-first list, the even-index identity and the open-question context.
• Free and private: Everything runs locally in the browser; nothing is uploaded or stored.

1. Pick a mode — First N primes for a fixed-size list, or Index limit to sweep an entire range of indexes.
2. Choose the count or the limit — up to 25 primes, or an index ceiling such as 500.
3. Read the rows — each prime shows its Fibonacci index, its exact value and its digit count.
4. Use the summary — total primes found and the highest index reached.
5. Copy or download the values for analysis, tests or teaching material.

Example 1 — The start of everything. Generate the first 5 primes. The list is 2 (index 3), 3 (index 4), 5 (index 5), 13 (index 7), 89 (index 11) — the nearest analogues to the first five ordinary primes.

Example 2 — A famous jump. Generate the first 10 primes. After 233 and 1597 the list reaches 28657 at index 23 — note how the gaps between primes grow as the sequence accelerates.

Example 3 — Verify a conjecture locally. Use index mode with limit 100. The sweep finds 12 primes ending at F(83) = 99194853094755497, a 17-digit value proven prime by the in-browser test.

Example 4 — A 200-digit proof of life. Use index mode with limit 1000. The run resurfaces the known giants such as F(569) and F(571), each over a hundred digits, each verified exactly.

Example 5 — Classroom reproducibility. Generate the first 8 twice. Both runs print 2, 3, 5, 13, 89, 233, 1597, 28657 — identical order, identical values, no randomness anywhere.

• Exact, not approximate: BigInt values up to 200+ digits with complete precision.
• Fast index screening: The even-index identity eliminates most candidates before any primality test runs.
• Trustworthy primality: Deterministic Miller-Rabin bases for the entire practical range of the tool.
• Complete metadata: Index and digit count per prime make every row independently verifiable.
• Reproducible output: Fixed lists, identical every run, ready for tests and teaching.
• Private and free: All computation happens in the browser; nothing is uploaded, stored or logged.