Divisibility properties of a Fibonacci-like sequence

Some Divisibility Properties of Generalized Fibonacci Sequences

Let c be any square-free integer, p any odd prime such that (c/p) = -1, and n any positive integer. The quantity ./IT, which would ordinarily be defined (mod p) as one of the two solutions of the congruence: x E c (mod p n ) , does not exist. Nevertheless, we may deal with objects of the form a + b/c~(mod p), where a and b are integers, in much the same way that we deal with complex numbers, th...

A Fibonacci-like Sequence of Composite Numbers

In 1964, Ronald Graham proved that there exist relatively prime natural numbers a and b such that the sequence {An} defined by An = An−1 +An−2 (n ≥ 2;A0 = a,A1 = b) contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth’s computation, Herbert ...

Properties of Periodic Fibonacci-like Sequences

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence . It is shown that this sequence is periodic with the period ( ) if | | .

A New Kind of Fibonacci-Like Sequence of Composite Numbers

An integer sequence (xn)n≥0 is said to be Fibonacci-like if it satisfies the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. We construct a new type of Fibonacci-like sequence of composite numbers. 1 The problem and previous results In this paper we consider Fibonacci-like sequences, that is, sequences (xn) ∞ n=0 satisfying the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. (1)

Divisibility by Fibonacci and Lucas Squares