Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.7 |

Department of Mathematics

Indian Institute of Technology Patna

Patliputra Colony, Patna - 800013

India

**Abstract:**

Let, as usual, *F**n* and *L**n*
denote the *n*th Fibonacci number and the *n*th
Lucas number, respectively. In this paper, we consider the Fibonacci
numbers *F*_{2}, *F*_{3}, ..., *F*_{t}.
Let *n* ≥ 1 be an integer such that 4*n*+2 ≤
*t* ≤ 4*n*+5 and *m* = *F*_{2n+2} +
*F*_{2n+4} = *L*_{2n+3}.
We prove that
the integers *F*_{2} *F*_{2n+2,} *F*_{3}*F*_{2n+2}, . . . , *F*_{t}*F*_{2n+2} modulo *m* all belong to
the interval [*F*_{2n+1}, 3*F*_{2n+2}]. Furthermore, the endpoints of the
interval [*F*_{2n+1}, 3*F*_{2n+2}] are obtained only by the integers *F*_{4}*F*_{2n+2} and
*F*_{4n+2}*F*_{2n+2}, respectively.

(Concerned with sequence A000045.)

Received September 28 2012;
revised version received January 21 2013.
Published in *Journal of Integer Sequences*, January 26 2013.

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