Lucas numbers are a generalization of Fibonacci numbers. Let X²-PX+Q be a polynomial in X where P and Q are not zero. Let a and b be the solutions. We assume D = P² - 4Q does not equal zero. Then we define the Lucas numbers as

          U_n(P,Q) = (aⁿ-bⁿ)/(a-b)
          V_n(P,Q) = aⁿ+bⁿ

This means U₀(P,Q) = 0 and U₁(P,Q) = 1. V₀(P,Q) = 2 and V₁(P,Q)=P.

Fibonacci numbers come from the special case U_n(1,-1) which produce the Fibonacci numbers.
          1,1,2,3,5,8,13...
The Fibonacci numbers obey the law F_n = F_n-1+F_n-2

For the Lucas numbers there are similar recurrence formulas:

          U_n(P,Q) = P U_n-1(P,Q) - Q U_n-2(P,Q)
          V_n(P,Q) = P V_n-1(P,Q) - Q V_n-2(P,Q)

If P=3 Q=2 then

          U_n(3,2) = 2ⁿ - 1
          V_n(3,2) = 2ⁿ + 1

As can be seen many sequences can be generated by Lucas numbers.

Lucas numbers can be used to test whether a number is prime. An example of those type of tests is the following:

Theorem

Let N > 1 be an odd integer and suppose that q_i are the prime factors of N+1. Assume that there exist a D such that D mod p does not equal a² mod p for any a, and for every q_i there exists a Lucas sequence U_(i) with P_i²-4 Q_i=D, where P_i,Q_i have no common prime factors or Q_i and N have no common prime factors and such that N divides U⁽ⁱ⁾_N+1 and N does not divide U⁽ⁱ⁾_N/qi. Then N is prime.

Below you can put in  a P and Q and generate the Nth Lucas numbers.

         P                                                    Q                                                 N

                                   

U                  V    

Fermat's Little Theorem
If p is a prime and a is an integer then a^p is congruent to a mod p.
Test it out:

p        a          a^p mod p                       a mod p                                                                   
                                                               

There is an analogue theorem for Lucas Numbers:

Theorem
Let p be an odd prime (not 2).
If p divides P and p divides Q then p divides U_N for every N>1.
If p divides P and p does not divide Q then p divides U_N exactly when  N is even.
If p does not divide P and p divides Q then p does not divide U_N for N ≥ 1.
If p does not divide P and p does not divide Q and p divides P²-4Q then p divides U_N exactly when p divides N.

Try it out below using the same P and Q and N as above:

 P               Q                 N                 U              P² - 4Q        
 

 p                                                                               

           


Disclaimer: There may be overflow because Lucas numbers get large fast. This will be indicated where U appears. It will say Overflow occured

Reference: All material about Lucas numbers is taken from The Little Book of Bigger Primes by Paulo Ribenboim. There is a lot more about Lucas numbers in the book. It's a fun book about primes though its heavy on the mathematics.