On the Frobenius number of a modular Diophantine inequality

José Carlos Rosales; P. Vasco

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 4, page 367-375
  • ISSN: 0862-7959

Abstract

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We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality a x mod b x , with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

How to cite

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Rosales, José Carlos, and Vasco, P.. "On the Frobenius number of a modular Diophantine inequality." Mathematica Bohemica 133.4 (2008): 367-375. <http://eudml.org/doc/250542>.

@article{Rosales2008,
abstract = {We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality $ax ~\@mod \;b\le x$, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.},
author = {Rosales, José Carlos, Vasco, P.},
journal = {Mathematica Bohemica},
keywords = {numerical semigroup; Diophantine inequality; Frobenius number; multiplicity; numerical semigroup; Diophantine inequality; Frobenius number; multiplicity},
language = {eng},
number = {4},
pages = {367-375},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Frobenius number of a modular Diophantine inequality},
url = {http://eudml.org/doc/250542},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Rosales, José Carlos
AU - Vasco, P.
TI - On the Frobenius number of a modular Diophantine inequality
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 4
SP - 367
EP - 375
AB - We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality $ax ~\@mod \;b\le x$, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.
LA - eng
KW - numerical semigroup; Diophantine inequality; Frobenius number; multiplicity; numerical semigroup; Diophantine inequality; Frobenius number; multiplicity
UR - http://eudml.org/doc/250542
ER -

References

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  1. Barucci, V., Dobbs, D. E., Fontana, M., Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains, Memoirs of the Amer. Math. Soc. 598 (1997). (1997) Zbl0868.13003MR1357822
  2. Delgado, M., Rosales, J. C., On the Frobenius number of a proportionally modular Diophantine inequality, Portugaliae Mathematica 63 (2006), 415-425. (2006) Zbl1172.11011MR2287275
  3. Alfonsín, J. L. Ramírez, The Diophantine Frobenius Problem, Oxford Univ. Press (2005). (2005) MR2260521
  4. Rosales, J. C., Modular Diophantine inequalities and some of their invariants, Indian J. Pure App. Math. 36 (2005), 417-429. (2005) Zbl1094.20037MR2199217
  5. Rosales, J. C., García-Sánchez, P. A., Finitely Generated Commutative Monoids, Nova Science Publishers, New York (1999). (1999) MR1694173
  6. Rosales, J. C., García-Sánchez, P. A., a-García, J. I. Garcí, Urbano-Blanco, J. M., 10.1016/j.jnt.2003.06.002, J. Number Theory 103 (2003), 281-294. (2003) MR2020273DOI10.1016/j.jnt.2003.06.002
  7. Rosales, J. C., García-Sánchez, P. A., Urbano-Blanco, J. M., 10.2140/pjm.2005.218.379, Pacific J. Math. 218 (2005), 379-398. (2005) Zbl1184.20052MR2218353DOI10.2140/pjm.2005.218.379
  8. Rosales, J. C., Urbano-Blanco, J. M., 10.1016/j.jalgebra.2006.08.009, J. Algebra 306 (2006), 368-377. (2006) Zbl1109.20052MR2271340DOI10.1016/j.jalgebra.2006.08.009
  9. Rosales, J. C., Vasco, P., The smallest positive integer that is solution of a proportionally modular Diophantine inequality, Math. Inequal. Appl. 11 (2008), 203-212. (2008) Zbl1142.20042MR2410272

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