The Formula of the Frobenius Number for a Numerical Semigroup with Embedding Dimension Three respect to a Partial Order Relation

Pitiwat Lueangwitchajaroen; Ekkachai Laysirikul

doi:10.14456/nujst.2023.13

Authors

Pitiwat Lueangwitchajaroen Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000
Ekkachai Laysirikul Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000

DOI:

https://doi.org/10.14456/nujst.2023.13

Keywords:

Embedding dimension, Frobenius number, Genus number, Numerical semigroup, Partial order relation

Abstract

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.

References

Barucci, V., Dobbs, D. E., & Fontana, M. (1997). Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Memoirs of the American Mathematical Society, 125(598), 78.

Curtis, F. (1990). On formulas for the Frobenius number of a numerical semigroup. Mathematica Scandinavica, 67, 190.

Davison, J. L. (1994). On the linear diophantine of Frobenius. Journal of Number Theory, 48, 353–363.

Froberg, R., Gottlieb, G., & Haggkvist, R. (1987). On numerical semigroups. Semigroup Forum, 35, 63–83.

Gu, Z. (2020). On the numerical semigroup generated by {b^(n+1+i)+b^(n+i)/(b-1) | i∈N}. Discrete Mathematics and Applications, 30, 257–264.

Gu, Z., Li, D., & Huang, Y. (2022). On the numerical semigroup generated by {2k-2mn+i-1 | i∈N}. Journal of Mathematics, 2022, 1–9.

Johnson, S. M. (1960). A linear diophantine problem. Canadian Journal of Mathematics, 12, 390-398.

Nari, H., Numata, T., & Watanabe, K. (2012). Genus of numerical semigroups generated by three elements. Journal of Algebra, 358, 67—73.

Ramirez-Alfonsin, J. L. (2005). The Diophantine Frobenius Problem. Oxford University Press, Oxford.

Rosales, J. C., & Garcia-Sanchez, P. A. (2009). Numerical semigroups. New York: Springer.

Selmer, E. S., & Beyer, O. (1978). On the linear diophantine problem of Frobenius in three variables. The Journal für die reine und angewandte Mathematik, 301, 161–170.

Sylvester, J. J. (1884). Mathematical questions with their solutions. Educational Times, 41, 21.

Tripathi, A. (2017). Formulae for the frobenius number in three variables. Journal of Number Theory, 170, 368-389.