The Formula of the Frobenius Number for a Numerical Semigroup with Embedding Dimension Three respect to a Partial Order Relation - Lueangwitchajaroen, Pitiwat; Laysirikul, Ekkachai (Naresuan University Journal: Science and Technology (NUJST); Vol 31 No 2 (2023): April-June 2023, 2023)
The Frobenius number is the largest positive integer that cannot be expressed as a non - negative linear combination of a given set of positive integers . It is considered to be one of the well - known problems in number theory, especially where the cardinality of the set is greater than three . Recently, algorithms and formulas have been proposed to calculate the Frobenius number in three variables, and various techniques have been used to handle the problem . Currently, for more than three variables, the problem associated with finding the Frobenius number is still considered to be an open problem . In this work, we used the concept of a numerical semigroup to develop an alternative approach to finding the Frobenius and genus number in three variables in particular cases . In the arbitrary variables, the formula presented in three variables can yield an upper bound of the Frobenius number and genus number. Keywords : Embedding dimension, Frobenius number, Genus number, Numerical semigroup, Partial order relation