ON PELL NUMBERS OF THE FORM PXj WHERE P IS PRIME
نویسنده
چکیده
(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 sake of convenience, we occasionally write u(n) instead of un and P(n) instead of Pn. Table 1, below, lists Pn and Rn for 1 < rc < 50.
منابع مشابه
Cullen Numbers in Binary Recurrent Sequences
In this paper, we prove that the generalized Cullen numbers, Cn(s, l) := sn ·2+ l, where l is an integer and s := (sn)n≥0 is a sequence of integers satisfying log |sn| < 2n−1 + O(1), occur only finitely many times in binary recurrent sequences (un)n≥0 whose characteristic roots are quadratic units and that satisfy some additional conditions. We also generalize this result in some sense to show ...
متن کاملON FIBONACCI NUMBERS OF THE FORM PXj WHERE P IS PRIME
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...
متن کاملOn the sum of Pell and Jacobsthal numbers by matrix method
In this paper, we define two $n$-square upper Hessenberg matrices one of which corresponds to the adjacency matrix of a directed pseudo graph. We investigate relations between permanents and determinants of these upper Hessenberg matrices, and sum formulas of the well-known Pell and Jacobsthal sequences. Finally, we present two Maple 13 procedures in order to calculate permanents of t...
متن کاملOn Some Diophantine Equations (i)
In this paper we study the equation m−n = py,where p is a prime natural number, p≥ 3. Using the above result, we study the equations x + 6pxy + py = z and the equations ck(x 4 + 6pxy + py) + 4pdk(x y + pxy) = z, where the prime number p ∈ {3, 7, 11, 19} and (ck, dk) is a solution of the Pell equation, either of the form c −pd = 1 or of the form c − pd = −1. I. Preliminaries. We recall some nece...
متن کاملOn The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices
In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de ned by the recurrence...
متن کامل