Generate Lucas Primes

What is a Lucas Prime?

A Lucas Prime is a Lucas number that is also a prime number. The Lucas sequence ($L_n$) is defined by starting with $L_0 = 2$ and $L_1 = 1$, where each subsequent number is the sum of the two preceding ones:

$$L_0 = 2,\quad L_1 = 1$$ $$L_n = L_{n-1} + L_{n-2} \quad \text{for } n \ge 2$$

A prime number is a positive integer greater than $1$ that has no positive divisors other than $1$ and itself. Looking at the first few Lucas numbers: $$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, \dots$$ we can identify the initial prime values:

$L_0 = 2$ (prime)
$L_2 = 3$ (prime)
$L_4 = 7$ (prime)
$L_5 = 11$ (prime)
$L_7 = 29$ (prime)
$L_8 = 47$ (prime)
$L_{11} = 199$ (prime)
$L_{13} = 521$ (prime)
$L_{16} = 2207$ (prime)
$L_{17} = 3571$ (prime)

Mathematical Properties of Lucas Primes

Lucas primes are of great interest in number theory and possess unique characteristics:

Indices Rule: If $L_n$ is a prime number, then its index $n$ must be $0$, a prime number (like $2, 5, 7, 11$), or a power of $2$ (like $4, 8, 16$). No other composite index can produce a Lucas prime.
Powers of 2: Lucas numbers of the form $L_{2^m}$ are only known to be prime for $m = 1, 2, 3,$ and $4$ (indices $2, 4, 8,$ and $16$). It is an open question whether any larger power of 2 produces a prime.
Growth Rate: Lucas primes grow at an exponential rate. For instance, the 11th prime in the sequence ($L_{19}$) is $9,349$, the 20th ($L_{113}$) has 24 digits, and the 28th ($L_{1361}$) has a massive $285$ digits!

How to Use This Generator

This specialized generator lets you explore and format Lucas primes with ease:

Select the Generation Mode: Choose "First N Lucas Primes" to list terms sequentially, or choose "By Index Range n" to search within a specific index band (filtering the OEIS index list).
Configure the Number Format: Render the huge prime numbers in Decimal (Base 10), Hexadecimal (Base 16), Octal (Base 8), or Binary (Base 2).
Choose your preferred Output Separator: You can separate list elements using newlines, commas, spaces, semicolons, or custom string sequences.
Toggle Show L(n) indices to prefix results with their respective subscripts (e.g. $L(11) = 199$) or output the pure numbers.
Instantly export results using the quick Action buttons to copy to your clipboard or download them as a text file.

Frequently Asked Questions

What is the first Lucas prime?

The first Lucas prime is $2$, which occurs at index $0$ ($L_0 = 2$). The next is $3$ ($L_2 = 3$), followed by $7$ ($L_4 = 7$) and $11$ ($L_5 = 11$).

Does every prime index n produce a Lucas prime?

No. While the index $n$ (except for powers of 2) must be prime for $L_n$ to be prime, a prime index does not guarantee primality. For example, $3$ is a prime number, but $L_3 = 4$, which is composite. Similarly, $23$ is prime, but $L_{23} = 64079 = 139 \times 461$, which is composite.

How many Lucas primes are currently known?

There are only a few dozen confirmed Lucas primes because they grow so quickly that proving primality for very large indices requires massive computer clusters running specialized Lucas-Lehmer-like tests. This tool precomputes the indices up to $n = 1361$, which yields primes up to $285$ digits.