Fibonacci Number Test

Fibonacci numbers repeat so often in the wild — sunflower spirals, rabbit-problem textbooks, spreadsheet roundings — that "is this number Fibonacci?" comes up constantly. A quick glance works up to 21, but beyond that the sequence jumps by the golden ratio per step, and guessing whether 75025 or 121393 sits in the list is exactly the kind of thing eyeballs get wrong.

Fibonacci Number Test decides definitively using the classic Lucas identity: n is Fibonacci exactly when 5n squared plus 4 or 5n squared minus 4 is a perfect square. BigInt square roots make the check exact at any size, a confirmed number gets its exact index, and a rejected one gets its Fibonacci neighbors so you can see where it actually sits. Batches of 10,000 values test in one pass, and everything runs in your browser with nothing uploaded.

• Definitive verdict: The 5n squared ± 4 perfect-square identity, computed exactly with BigInt.
• Exact index: Confirmed numbers report k with F(k) = n, up to 209-digit values like F(1000).
• Neighbor display: Rejected values show the Fibonacci numbers immediately below and above.
• Special cases handled: 0 is F(0); 1 is both F(1) and F(2); negatives are never Fibonacci.
• Batch testing: Up to 10,000 values, each verdict independent with full detail.
• No rounding anywhere: Square roots run on BigInt — exact for hundreds of digits.
• Copy or download: Take the verdict list to the clipboard or save it as plain text.
• Reference tab: The identity, the first terms, and the golden-ratio growth story.
• Free and private: All computation happens 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 each verdict — YES with its index, or NO with its Fibonacci neighbors.
3. Inspect the row detail — index, position and bracketing neighbors for every value.
4. Check the identity note — which of the two square tests resolved the value.
5. Copy or download the results for spreadsheets, tests or teaching.

Example 1 — The famous small truths. Enter 8 and 55. Both return YES — 8 is F(6) and 55 is F(10), and the sequence rolls on without a break.

Example 2 — A five-digit survivor. Enter 75025. The tool reports YES at index 25, confirming a value that hand-listing would make you doubt.

Example 3 — Near miss. Enter 1000. The verdict is NO, with neighbors 987 (F(16)) and 1597 (F(17)) — the names on either side of the gap, ready for comparison.

Example 4 — A giant. Enter 354224848179261915075. The tool says YES at index 100 — the twentieth Fibonacci number, 21 digits, decided exactly by the square test.

Example 5 — The double count. Enter 1. The tool explains the special position: F(1) and F(2) are both this value — the only duplicated term in the whole sequence.

• Exact at any size: BigInt squares and roots — no rounding, no false positives.
• Index included: One number answers both questions: membership and position.
• Neighbors for context: Near misses show exactly where the value falls in the golden-ratio ladder.
• Batch ready: Ten thousand values in one pass with per-line verdicts.
• Educational by design: The identity and neighbor columns teach the structure, not just the answer.
• Private and free: Everything runs in the browser; nothing is uploaded, stored or logged.