OJDM  Vol.9 No.1 , January 2019
On the Modular Erdös-Burgess Constant
Abstract: Let n be a positive integer. For any integer a, we say that is idempotent modulo n if a2≡a(mod n). The n-modular Erdös-Burgess constant is the smallest positive integer l such that any l integers contain one or more integers, whose product is idempotent modulo n. We gave a sharp lower bound of the n-modular Erdös-Burgess constant, in particular, we determined the n-modular Erdös-Burgess constant in the case when n was a prime power or a product of pairwise distinct primes.
Cite this paper: Hao, J. , Wang, H. and Zhang, L. (2019) On the Modular Erdös-Burgess Constant. Open Journal of Discrete Mathematics, 9, 11-16. doi: 10.4236/ojdm.2019.91003.

[1]   Gao, W. and Geroldinger, A. (2006) Zero-Sum Problems in Finite Abelian Groups: A Survey. Expositiones Mathematicae, 24, 337-369.

[2]   Geroldinger, A. and Halter-Koch, F. (2006) Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Pure Appl. Math., Vol. 278, Chapman & Hall/CRC.

[3]   Geroldinger, A. and Ruzsa, I. (2009) Combinatorial Number Theory and Additive Group Theory. In Advanced Courses in Mathematics CRM Barcelona, Springer, Birkhauser.

[4]   Grynkiewicz, D.J. (2013) Structural Additive Theory, Developments in Mathematics. Vol. 30, Springer, Cham.

[5]   Tao, T. and Van Vu, H. (2006) Additive Combinatorics. Cambridge University Press, Cambridge.

[6]   Cziszter, K., Domokos, M. and Geroldinger, A. (2016) The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics, Multiplicative Ideal Theory and Factorization Theory. Springer, Berlin, 43-95.

[7]   Gao, W., Li, Y. and Peng, J. (2014) An Upper Bound for the Davenport Constant of Finite Groups. Journal of Pure and Applied Algebra, 218, 1838-1844.

[8]   Wang, G. (2015) Davenport Constant for Semigroups II. Journal of Number Theory, 153, 124-134.

[9]   Wang, G. (2017) Additively Irreducible Sequences in Commutative Semigroups. Journal of Combinatorial Theory, Series A, 152, 380-397.

[10]   Wang, G. and Gao, W. (2008) Davenport Constant for Semigroups. Semigroup Forum, 76, 234-238.

[11]   Wang, G. and Gao, W. (2016) Davenport Constant of the Multiplicative Semigroup of the Ring . arXiv:1603.06030

[12]   Zhang, L., Wang, H. and Qu, Y. (2017) A Problem of Wang on Davenport Constant for the Multiplicative Semigroup of the Quotient Ring of . Colloquium Mathematicum, 148, 123-130.

[13]   Burgess, D.A. (1969) A Problem on Semi-Groups. Studia Sci. Math. Hungar., 4, 9-11.

[14]   Gillam, D.W.H., Hall, T.E. and Williams, N.H. (1972) On Finite Semigroups and Idempotents. Bulletin of the London Mathematical Society, 4, 143-144.

[15]   Wang, G. (2019) Structure of the Largest Idempotent-Product Free Sequences in Semigroups. Journal of Number Theory, 195, 84-95.

[16]   Wang, G. (2018) Erdos-Burgess Constant of the Direct Product of Cyclic Semigroups. arXiv:1802.08791.

[17]   Wang, H., Hao, J. and Zhang, L. (2018) On the Erdos-Burgess Constant of the Multiplicative Semigroup of a Factor Ring of . International Journal of Number Theory. (To Appear)

[18]   Grynkiewicz, D.J. (2013) The Large Davenport Constant II: General Upper Bounds. Journal of Pure and Applied Algebra, 217, 2221-2246.