Lucas Prime Test

A Lucas prime must satisfy two separate questions at once: does it belong to the Lucas sequence 2, 1, 3, 4, 7, 11, 18, 29, 47... and does it have only one and itself as divisors? Small answers are easy — 7, 47, 199 — but traps wait everywhere: 18 is Lucas and not prime, 29 is both, 5 is prime and never Lucas, and even 2207, the ninth Lucas prime, fools the unaided eye. Membership alone never decides anything.

Lucas Prime Test runs both checks exactly. The Lucas square identity — 5n squared ± 20 a perfect square — decides membership and the index; a twelve-base Miller-Rabin test decides primality on BigInt values, immune to rounding. Verdicts come as one of four labels, batches of 10,000 values run in a single pass, and nothing ever leaves the browser.

• Four exact verdicts: Lucas prime, Lucas composite, plain prime, or neither — never a vague answer.
• Exact membership: The Lucas square identity 5n² ± 20 with BigInt square roots.
• Exact primality: Miller-Rabin, twelve prime bases, deterministic below 3.3 x 10^24.
• Index retrieval: Every Lucas verdict reports its k, from L(4) = 7 to L(16) = 2207 and beyond.
• Special cases: 1 is L(1) yet not prime; 0 is not Lucas at all — both handled explicitly.
• Batch testing: Up to 10,000 values per run, each with independent verdict and detail.
• Copy or download: Verdict list and numbers list, both one-click exports.
• Reference tab: Known Lucas primes, the filters, the Fibonacci connection.
• 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 verdict badge — Lucas prime, Lucas composite, prime, or neither.
3. See the index — Lucas values carry their exact k with L(k) = n.
4. Follow the detail — which square candidate passed and which primality path decided.
5. Copy or download the results for research, teaching or further analysis.

Example 1 — The first answers. Enter 7, 47 and 199. All three return Lucas prime at indexes 4, 8 and 11 — three of the first nine Lucas primes.

Example 2 — The traps. Enter 18 and 123. Both are Lucas but composite: 18 = 2 x 3 x 3 and 123 = 3 x 41. The badge keeps membership and primality visibly separate.

Example 3 — The other side. Enter 5 and 29. 5 is prime but never Lucas; 29 is L(7) and prime — two verdicts that demand both tests.

Example 4 — A big verified jump. Enter 3010349. The tool reports Lucas prime at index 31 — a seven-digit value decided in milliseconds by the in-browser Miller-Rabin test.

Example 5 — Batch audit. Paste 2, 3, 5, 8, 18, 29, 47, 123, 199 together: the output mixes Lucas primes (2, 3, 29, 47, 199), Lucas composites (8, 18, 123) and the plain prime 5, exercising every verdict in one pass.

• Both tests, exact: Membership and primality are each proven, never guessed.
• Lucas-verified fast: The square identity answers membership instantly, so the index arrives with the verdict.
• Practical certainty: Deterministic Miller-Rabin bases cover the entire range ordinary numbers occupy.
• Educational clarity: Indexes and explanation rows show exactly why each verdict holds.
• 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.