Lucas Prime

Last Updated : 11 Oct, 2024

Before Discussing Lucas Prime let us understand Lucas numbers first. Lucas Numbers are a sequence of numbers similar to the Fibonacci sequence. Each Lucas number is generated by adding the two previous numbers in the sequence, just like Fibonacci numbers, but they start with different initial values i.e., L₀ = 2 and L₁ = 1.

From there, each following number is the sum of the previous two:

L₂ = L₀ + L₁ = 2 + 1 = 3
L₃ = L₁ + L₂ = 1 + 3 = 4
L₄ = L₂ + L₃ = 3 + 4 = 7
L₅ = L₃ + L₄ = 4 + 7 = 11
And so on.

First Few Lucas Numbers are:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, . . .

Lucas Prime

Lucas primes are a subset of prime numbers that are derived from the Lucas sequence. A Lucas prime is a Lucas number, that is also a prime number.

The first few Lucas Primes are:

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521, . . .

In the sequence of Lucas numbers, L_n+2 = L_n+1 + L_n. All values of n = 0, , 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849, . . .; are primes.

Note: largest known Lucas probable prime is L_1051849, which has 219824 decimal digits.

Properties of Lucas Prime

Some of the important properties of Lucas primes are:

As the sequence progresses, the ratio between consecutive Lucas numbers approaches the Golden Ratio. Specifically:
- \lim_{n \to \infty} \frac{L_n}{L_{n-1}} = \varphi

Lucas numbers satisfy an interesting identity involving Fibonacci numbers:
- L_n = F_n-1 + F_n+1 [Where Fn is the nth term of Fibonacci sequence.
- This shows that Lucas numbers can be expressed as the sum of two consecutive Fibonacci numbers.

Generating function for the Lucas sequence is:
- G(x) = \frac{2 - x}{1 - x - x^2}
- This function can be used to find any Lucas number in the sequence.

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