• Song, Kyunghwan (Institute of Mathematical Sciences Ewha Womans University)
  • Received : 2019.04.15
  • Accepted : 2019.12.18
  • Published : 2020.05.31


The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers.


  1. I. M. Aliev and P. M. Gruber, An optimal lower bound for the Frobenius problem, J. Number Theory 123 (2007), no. 1, 71-79.
  2. D. Beihoffer, J. Hendry, A. Nijenhuis, and S. Wagon, Faster algorithms for Frobenius numbers, Electron. J. Combin. 12 (2005), Research Paper 27, 38 pp.
  3. S. Bocker and Z. Liptak, A fast and simple algorithm for the money changing problem, Algorithmica 48 (2007), no. 4, 413-432.
  4. M. Bras-Amoros, Bounds on the number of numerical semigroups of a given genus, J. Pure Appl. Algebra 213 (2009), no. 6, 997-1001.
  5. A. Brauer and J. E. Shockley, On a problem of Frobenius, J. Reine Angew. Math. 211 (1962), 215-220.
  6. F. Curtis, On formulas for the Frobenius number of a numerical semigroup, Math. Scand. 67 (1990), no. 2, 190-192.
  7. L. G. Fel, On Frobenius numbers for symmetric (not complete intersection) semigroups generated by four elements, Semigroup Forum 93 (2016), no. 2, 423-426.
  8. B. K. Gil et al., Frobenius numbers of Pythagorean triples, Int. J. Number Theory 11 (2015), no. 2, 613-619.
  9. Z. Gu and X. Tang, The Frobenius problem for a class of numerical semigroups, Int. J. Number Theory 13 (2017), no. 5, 1335-1347.
  10. B. R. Heap and M. S. Lynn, On a linear Diophantine problem of Frobenius: An improved algorithm, Numer. Math. 7 (1965), 226-231.
  11. M. Hujter and B. Vizvari, The exact solutions to the Frobenius problem with three variables, J. Ramanujan Math. Soc. 2 (1987), no. 2, 117-143.
  12. M. Lepilov, J. O'Rourke, and I. Swanson, Frobenius numbers of numerical semigroups generated by three consecutive squares or cubes, Semigroup Forum 91 (2015), no. 1, 238-259.
  13. J. M. Marin, J. L. Ramirez Alfonsin, and M. P. Revuelta, On the Frobenius number of Fibonacci numerical semigroups, Integers 7 (2007), A14, 7 pp.
  14. G. Marquez-Campos, I. Ojeda, and J. M. Tornero, On the computation of the Apery set of numerical monoids and affine semigroups, Semigroup Forum 91 (2015), no. 1, 139-158.
  15. D. C. Ong and V. Ponomarenko, The Frobenius number of geometric sequences, Integers 8 (2008), A33, 3 pp.
  16. R. W. Owens, An algorithm to solve the Frobenius problem, Math. Mag. 76 (2003), no. 4, 264-275.
  17. M. Raczunas and P. Chrzastowski-Wachtel, A Diophantine problem of Frobenius in terms of the least common multiple, Discrete Math. 150 (1996), no. 1-3, 347-357.
  18. J. L. Ramirez-Alfonsin, Complexity of the Frobenius problem, Combinatorica 16 (1996), no. 1, 143-147.
  19. A. M. Robles-Perez and J. C. Rosales, The Frobenius problem for numerical semigroups with embedding dimension equal to three, Math. Comp. 81 (2012), no. 279, 1609-1617.
  20. J. C. Rosales, Numerical semigroups with Apery sets of unique expression, J. Algebra 226 (2000), no. 1, 479-487.
  21. J. C. Rosales, M. B. Branco, and D. Torrao, The Frobenius problem for Thabit numerical semigroups, J. Number Theory 155 (2015), 85-99.
  22. J. C. Rosales, M. B. Branco, and D. Torrao, The Frobenius problem for repunit numerical semigroups, Ramanujan J. 40 (2016), no. 2, 323-334.
  23. J. C. Rosales, M. B. Branco, and D. Torrao, The Frobenius problem for Mersenne numerical semigroups, Math. Z. 286 (2017), no. 1-2, 741-749.
  24. J. C. Rosales and P. A. Garcia-Sanchez, Numerical semigroups, Developments in Mathematics, 20, Springer, New York, 2009.
  25. J. C. Rosales, P. A. Garcia-Sancheza, J. I. Garcia-Garcia, and J. A. Jimenez Madridb, Fundamental gaps in numerical semigroups, J. Pure Appl. Algebra 189 (2004), no. 1-3, 301-313.
  26. J. L. Ramirez Alfonsin and O. J. Rdseth, Numerical semigroups: Apery sets and Hilbert series, Semigroup Forum 79 (2009), no. 2, 323-340. 009-9133-5
  27. E. S. Selmer, On the linear Diophantine problem of Frobenius, J. Reine Angew. Math. 293(294) (1977), 1-17.
  28. V. Shchur, Ya. Sinai, and A. Ustinov, Limiting distribution of Frobenius numbers for n = 3, J. Number Theory 129 (2009), no. 11, 2778-2789.
  29. J. J. Sylvester, Problem 7382, The Educational Times, and Journal of College Of Preceptors, New Series 36 (1883), no. 266, 177.
  30. K. H. Song, The Frobenius problem for extended Thabit numerical semigroups, Preprint.
  31. A. Tripathi, Formulae for the Frobenius number in three variables, J. Number Theory 170 (2017), 368-389.