The p-Frobenius and p-Sylvester numbers for Fibonacci and Lucas triplets

Abstract

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let a1,a2,…,al be positive integers such that their greatest common divisor is one. For a nonnegative integer p, denote the p-Frobenius number by gp(a1,a2,…,al), which is the largest integer that can be represented at most p ways by a linear combination with nonnegative integer coefficients of a1,a2,…,al. When p=0, the 0-Frobenius number is the classical Frobenius number. When l=2, the p-Frobenius number is explicitly given. However, when l=3 and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when p>0, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where l=3. In this paper, we show the explicit formula for the Fibonacci triple when p>0. In addition, we give an explicit formula for the p-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most p ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

Keywords

• linear diophantine problem of frobenius
• frobenius numbers
• sylvester numbers
• the number of representations
• apéry set
• fibonacci numbers