On the Frobenius number for three variables

Abstract

For positive integers $a$, $b$, and $c$ which have no common divisor, the Frobenius number of $a$, $b$ and $c$ is defined to be the largest integer that cannot be expressed as a linear combination of $a$, $b$ and $c$ with non-negative integer coefficients. In 2017, Tripathi gave an algorithmic formula for the Frobenius number in three variables, however there were some minor inconsistencies in the formula. In this paper, we settle these inconsistencies.

Figures (2)

• Figure 1: An example of a graphical depiction of the values in $\mathscr{X}$ where the slope (depicted by the green line) going through the lowest points has a negative gradient. The red dots denote the $x$-values less than $u$.
• Figure 2: An example of a graphical depiction of the values in $\mathscr{X}$ where the slope (see the green lines) going through the lowest points has a positive gradient. The height of the green and pink lines represent the distances $d_1 = \widehat{x}$ and $d_2 = \widehat{x}+u-x_\mu$ respectively.

Theorems & Definitions (41)

• Theorem 2.1
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Remark 2.8
• Remark 2.10