Lucas Primes

Number Theory Level 1

Lucas numbers are similar to Fibonacci numbers , each number is defined to be the sum of its two previous terms. The first two Lucas numbers are $2, 1$ as opposed to the first two Fibonacci numbers $0, 1$ . (Assume the sequence is extended with initial term $n = 0$ ).

The sequence of Lucas numbers is

$2, 1, 3, 4, 7, 11, 18, 29, 47, 76...$

How many prime Lucas numbers less than $10000$ are there?

The answer is 11.

1 solution

Wind Quotidian
Dec 6, 2017

I did this by simply calculating the sequence that comes after.

The first few Lucas numbers are as follows.

$0 = 2$

$1 = 1$

$2 = 3$

$3 = 4 = 2X2$

$4 = 7$

$5 = 11$

$6 = 18 = 2X2X3$

$7 = 29$

$8 = 47$

$9 = 76 = 2X2X19$

$10 = 123 = 3X41$

$11 = 199$

$12 = 322 = 2X7X23$

$13 = 521$

$14 = 843 = 3X281$

$15 = 1364 = 22X11X31$

$16 = 2207$

$17 = 3571$

$18 = 5778 = 2X33X107$

$19 = 9349$

$20 = 15127 = 7X2161$

As we can see, the $20th$ term exceeds $10000$ , therefore we can check whether the numbers before the $20th$ term are prime.

Lucas Primes

The answer is 11.

1 solution

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