Fibonacci Primitive Roots and the Period of the Fibonacci Numbers Modulop
نویسنده
چکیده
One says g is a Fibonacci primitive root modulo /?, wherep is a prime, iff g is a primitive root modulo/7 and g = g + 1 (mod p). In [1 ] , [2 ] , and [3] some interesting properties of Fibonacci primitive roots were developed. In this paper, we shall show that a necessary and sufficient condition for a prime/? ^ 5 to have a Fibonacci primitive root is p = 1 or 9 (mod 10) and Alp) = p 1, where/I//?,/ is the period of the Fibonacci numbers modulo p (Theorem 1); for/? = 11 or 19 (mod 20), we shall explicitly determine the Fibonacci primitive root if it exists (Proposition 1). In the sequel, Fn will denote the/7 Fibonacci number and/? will denote a prime greater than five.
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