The Fibonacci Triangles And Lucas Triangles

The Fibonacci numbers are defined recurrently by F_{n + 2} = F_n+1+F_n with F₀ = 0, F₁=1, where n≥0. The Lucas numbers are defined recurrently by L_{n + 2} = L_n+1+L_n with L₀ = 2, L₁=1, where n≥0. Fibonacci sequences come from the rabbit problem posed by an Italian mathematician Fibonacci in the 12th century. It is a very interesting sequence derived from a quite brief recurrence relation, and is closely related to the natural phenomena and the mathematical knowledge, such as geometric graphs, Golden Cut, Yang Hui Triangle, matrix operations. It is extensively applied to the fields of optimum seeking methods and computer science etc. The Fibonacci sequence and the Lucas sequence are important research contents in number theory.We study Fibonacci Triangles and Lucas Triangles in this paper by the property of Fibonacci sequences and Lucas sequences. H.Harborth and A.Kemnitz put forward Fibonacci Triangles conjecture in 1990.Later there are some researches in China. We give some proofs of the Fibonacci Triangles conjecture and solve completely the Lucas Triangles conjecture in this paper.