NOTE ON THE MATHEMATICAL BEAUTY OF LUCAS NUMBERS

The Lucas numbers are a sequence of integers defined by the linear recurrence equation. The values include., 123,76,47,29,18,11,7,4,3,1,2.. This sequence is closely related to the Fibonacci numbers and satisfies the same recurrence. As in the Fibonacci number sequence, each number is defined by the sum of the two immediate previous terms. The only square numbers found in the Lucas sequence are 1 and 4. The only triangular Lucas numbers are 1, 3, and 5778. The only cubic number found in this sequence is 1. A Lucas number may be defined as follows: L-n :={2 if n=0; 1 if n=1; Ln-1 + Ln-2 if n>1. In this paper I will investigate the properties of the Lucas numbers and discuss different patterns and theorems found within this series. I will compare and contrast the close relationship between the Fibonacci and Lucas numbers. Each theorem and pattern will be supplemented with different examples and proofs.