Lucas Primes Generator

The Lucas sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322... has a secret: sprinkled among those values are primes. L(4) = 7, L(5) = 11, L(7) = 29, L(8) = 47, then a long dry stretch until L(11) = 199 and L(13) = 521. The pattern is elusive — most Lucas numbers are composite, some are even, some are trapped divisible by 3 — and only a scanning test with exact arithmetic can separate the gold from the gravel.

Lucas Primes Generator scans the sequence from index 0 up to any bound you set, skips whole composite families with algebraic filters, and verifies every surviving candidate with a twelve-base Miller-Rabin test on BigInt values. The result is a clean list: index, full decimal value, verification note. Progress phases keep large scans responsive, and one click copies or downloads everything.

• Complete scan: Every prime in the Lucas sequence up to your chosen maximum index.
• Algebraic filters: Index families 2 mod 4 (divisible by 3) and multiples of 3 (even) skipped before any test.
• Exact primality: Miller-Rabin with twelve prime bases — deterministic for every size this tool produces in practice.
• BigInt values: Full decimal expansion of each prime, no rounding, no exponents.
• Phased progress: Large scans run in visible phases with live counters and a Stop button.
• Copy or download: Clean L(n) = value (prime) lines for reports, tests or datasets.
• Reference tab: The first Lucas primes, the divisibility shortcuts, and the open infinity question.
• Free and private: Everything runs in the browser — nothing is uploaded, stored or logged.

1. Set the maximum index — where to stop the scan (4 to 2000).
2. Run the scan — phases tick by with live progress; values get tested the moment their turn arrives.
3. Read the primes — each row shows the index, the full value and its verification note.
4. Compare with the reference — L(4) = 7, L(5) = 11, L(7) = 29 should match the first rows exactly.
5. Copy or download — the complete verified list for further work.

Example 1 — The seed scan. Max index 20. Expect exactly these six primes: L(4) = 7, L(5) = 11, L(7) = 29, L(8) = 47, L(11) = 199, L(13) = 521, L(16) = 2207, L(17) = 3571, L(19) = 9349 — nine primes, each verified by Miller-Rabin.

Example 2 — The long dry stretch. Scan to 30 and watch the gap: after L(19) = 9349 the next prime appears only at L(31) = 3010349, a 7-digit value separated from its predecessor by twelve composite indexes.

Example 3 — The filters at work. Max index 24. The scan skips L(6), L(10), L(14), L(18), L(22) as multiples of 3 and L(3), L(9), L(15), L(21), L(24) as even — proving most of the sequence dies without a single primality test.

Example 4 — A deep sweep. Max index 200 finds the primes at indexes 31, 37, 41, 53, 61, 71, 79, 113, 131, 151, 173... with values growing to 35 digits — all exact.

Example 5 — Double-check with the checker. Feed L(16) = 2207 into the Fibonacci-prime companion or the Lucas-number checker on this site: the verdict agrees, independent verification in one tab switch.

• Verified list: Every row is a provable prime — the filters and Miller-Rabin run exact checks, not guesses.
• Speed by algebra: Whole composite families are dismissed before testing, not one by one.
• Honest spans: The famous gaps between Lucas primes show up exactly as they are — no artificial padding.
• Workable output: Formatted rows and copy-ready lines serve papers, tests and datasets alike.
• Responsive scans: Phased execution keeps even 2000-index sweeps interactive.
• Private by design: Computation stays in the browser; nothing is uploaded or logged.