ON FIBONACCI NUMBERS OF THE FORM PXj WHERE P IS PRIME
نویسنده
چکیده
INTRODUCTION Let p denote a prime, n a natural number, Fin) the nth Fibonacci number. Consider the equation: F(n) = px, (*) In [3], J. H. E. Cohn proved that for p = 2, the only solutions of (*) are (i) n = 3, x = 1 and (ii) n = 6, x = 4. In [8], R. Steiner proved that for p = 3, the only solution of (*) is n 4, a: = 1. Call a solution of (*) trivial if x = 1 . In this article, we solve O ) for all odd p such that p E 3 (mod 4) or p < 10,000. 'Except for p = 3,001, all solutions obtained are trivial. Lin) denotes the nth Lucas number. Definition 1 z{p) is the Fibonacci entry point of p, that is, zip) = m±n{k : k > 0 and p\F(k)}. Definition 2 yip) is the least prime factor of zip),
منابع مشابه
ON PELL NUMBERS OF THE FORM PXj WHERE P IS PRIME
(iv) un = (a b)/(a b), yn = a + £. If i4 = £ = 1, then un, Vn are the Fibonacci and Lucas sequences, respectively. If A = 3 and 5 =-2, then un, t;n are the Mersenne and Fermat sequences, respectively. If A = 2 and 5 = 1 (so that Z? = 8), then un is called the Pell sequence (see [4, p. 187]), and is denoted Pn; vn may be called the secondary Pell sequence, and denoted i?n, following [7]. For the...
متن کاملAn application of Fibonacci numbers into infinite Toeplitz matrices
The main purpose of this paper is to define a new regular matrix by using Fibonacci numbers and to investigate its matrix domain in the classical sequence spaces $ell _{p},ell _{infty },c$ and $c_{0}$, where $1leq p
متن کاملFibonacci Primitive Roots and the Period of the Fibonacci Numbers Modulop
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...
متن کاملFibonacci numbers of the form p a ± p b Florian Luca
In this paper, we show that the diophantine equation Fn = p ± p has only finitely many positive integer solutions (n, p, a, b), where p is a prime number and max{a, b} ≥ 2.
متن کاملFibonacci numbers and Fermat ’ s last theorem
numbers. As applications we obtain a new formula for the Fibonacci quotient Fp−( 5 p )/p and a criterion for the relation p |F(p−1)/4 (if p ≡ 1 (mod 4)), where p 6= 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonac...
متن کامل