Skip to main content

Goldbach’s conjecture in max-algebra

Abstract

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Burkard RE, Butkovič P (2003) Finding all essential terms of a characteristic maxpolynomial. Discrete Appl Math 130:367–380. doi:10.1016/S0166-218X(03)00223-3

    Article  Google Scholar 

  2. Butkovič P (2010) Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London. doi:10.1007/978-1-84996-299-5

  3. Cuninghame-Green RA (1979) Minimax Algebra. Springer-Verlag, Berlin, Lecture Notes in Economics and Mathematical Systems

    Book  Google Scholar 

  4. Erdős P (1934) A Theorem of Sylvester and Schur. J Lond Math Soc 9:282–288. doi:10.1112/jlms/s1-9.4.282

    Article  Google Scholar 

  5. Heidergott B, Olsder GJ, van der Woude J (2006) Max plus at work. Modeling and analysis of synchronized systems. Princeton Series in Applied Mathematics. Princeton University Press, Princeton

  6. Heinig G (2001) Not every matrix is similar to a Toeplitz matrix. Linear Algebra Appl 332–334:519–531. doi:10.1016/S0024-3795(01)00308-1

    Article  Google Scholar 

  7. Karp RM (1978) A characterization of the minimum cycle mean in a digraph. Discrete Math 23:309–311. doi:10.1016/0012-365X(78)90011-0

    Article  Google Scholar 

  8. Landau HJ (1994) The inverse eigenvalue problem for real symmetric Toeplitz matrices. J Am Math Soc 7:749–767

    Article  Google Scholar 

  9. Szabó P (2009) A short note on the weighted sub-partition mean of integers. Oper Res Lett 37(5):356–358. doi:10.1016/j.orl.2009.04.003

    Article  Google Scholar 

  10. Szabó P (2013) An iterative algorithm for computing the cycle mean of a Toeplitz matrix in special form. Kybernetika 49(4):636–643. http://www.kybernetika.cz/content/2013/4/636

  11. Zimmermann K (1976) Extremální algebra (in Czech). Ekonomický ústav ČSAV, Praha

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peter Szabó.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Szabó, P. Goldbach’s conjecture in max-algebra. Comput Manag Sci 14, 81–89 (2017). https://doi.org/10.1007/s10287-016-0268-z

Download citation

Keywords

  • Max-algebra
  • Triangular Toeplitz matrix
  • Goldbach’s conjecture