Abstract
我們稱g為Fibonacci數列上的原根(modulo p),即g是一個原根而且對模p,g*g-g-1與0同餘.在此篇論文中我們討論Fibonacci數列上原根(modulop)的性質,也找出Fibonacci數列上原根(modulo p)存在的充份必要條件.進一步我們也得到:令g是一個Fibonacci數列上原根(modulo p.則K(p*p)=pK(p)若且惟若存在一個Fibonacci數列上的原根(modulo nth powerofp),n為大於1的正整數,K(p)指Fibonacci數列(modulo p)週期的長度.此外,Fibonacci數列上原根(modulo p)的個數相等於Fibonacci數列上原根(modulo nth power of p)的個數.D.D. Wall提問是否K(p*p)=K(p).Zhi-Hong Sun and Zhi-Wei Sun指出Wall's conjecture會導出Fermat's lasttheorem.在此我們得到Wall's conjecture與Fibonacci數列的原根(modulo nth power of p)存在間的關係.We say g is a Fibonacci primitive root modulo m if and only if gis a primitiveroot modulo m and g*g-g-1 is congruent to 0 modulom. In this paper, we obtainthe conditions of the existence ofFibonacci primitive roots modulo p where p an odd prime. We alsoget that let g be a Fibonacci primitive root modulo p, then K(p*p)=pK(p) if and only if there exists a Fibonacci primitive rootmodulo nth power of p where K(p) is the length of the period ofthe Fibonacci sequence modulo p and n is more than 2.Furthermore, the number of Fibonacci primitive roots modulo nthpower of p is the same as the number of Fibonacci primitiveroots modulo p.Here, we also have the relations between theWall's conjecture and the existence of Fibonacci primitive rootsmodulo nth power of p.