On Congruences For N-step Q-Lucas Numbers

The topic of congruences is one of the main parts of combinatorial number theory, which has been studied intensively by mathematicians in recent years. In this paper, we obtain a congruence on n-step q-Lucas numbers, and prove it by using combinatorial methods, then make a further extension.In Chapter 1, we give an introduction to n-step q-Fibonacci numbers and n-step q-Lucas numbers as well as related results. In particular, we focus on those results related to this paper. Meanwhile we present our main theorems in this chapter.In Chapter 2, we first introduce the combinatorial explanations of the n-step Fi-bonacci numbers Fk(n) and n-step Lucas numbers Lk(n) related to cyclic layered partitions of{1,2,…, k}, then we prove the main theorems in this paper.In Chapter 3, we extend our results to n-step q-Pell-Lucas numbers.