Arkiv för Matematik

, Volume 42, Issue 2, pp 301–306

# Every positive integer is the Frobenius number of an irreducible numerical semigroup with at most four generators

• Pedro A. García-Sánchez
• José C. Rosales
Article

## Abstract

Letg be a positive integer. We prove that there are positive integersn1,n2,n3 andn4 such that the semigroupS=(n1,n2,n3,n4) is an irreducible (symmetric or pseudosymmetric) numerical semigroup with g(S)=g.

## Keywords

Positive Integer Numerical Semigroup Frobenius Number Irreducible Numerical Semigroup
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Apéry, R., Sur les branches superlinéaires des courbes algébriques,C. R. Acad. Sci. Paris 222 (1946), 1198–1200.
2. 2.
Barucci, V., Dobbs, D. E. andFontana, M.,Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains, Mem. Amer. Math. Soc.125, Amer. Math. Soc., Providence, R. I., 1997.Google Scholar
3. 3.
Curtis, F., On formulas for the Frobenius number of a numerical semigroup,Math. Scand. 67 (1990), 190–192.
4. 4.
Fröberg, R., Gottlieb, C. andHäggkvist, R., On numerical semigroups,Semigroup Forum 35 (1987), 63–83.
5. 5.
Ramírez Alfonsín, J. L., The Diophantine Frobenius problemPreprint, Forschungsinstitut für Diskrete Mathematik, Bonn, 2000.Google Scholar
6. 6.
Rosales, J. C. andBranco, M. B., Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups,J. Pure Appl. Algebra 171 (2002), 303–314.
7. 7.
Rosales, J. C. andBranco, M. B., Decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups,Bull. Belg. Math. Soc. 9 (2002), 373–381.
8. 8.
Rosales, J. C. andBranco, M. B., Irreducible numerical semigroups,Pacific J. Math. 209 (2003), 131–143.
9. 9.
Rosales, J. C., García-Sánchez, P. A. andGarcía-García, J. I., Every positive integer is the Frobenius number of a numerical semigroup with three generators,Preprint, 2003.Google Scholar
10. 10.
Rosales, J. C., García-Sánchez, P. A., García-García, J. I. andJiménez-Madrid, J. A., The oversemigroups of a numerical semigroup,Semigroup Forum 67 (2003), 145–158.
11. 11.
Sylvester, J. J., Mathematical questions with their solutions,Educational Times 41 (1884), 21.