Abstract
In the year 1832, the well known German mathematician Carl Friedrich Gauss proposed the set of Gaussian integers, which corresponds to those complex numbers whose real and imaginary parts are integer numbers. A few years later, Gaussian primes were defined as Gaussian integers that are divisible only by its associated Gaussian integers. The Gaussian Moat problem asks if it is possible to walk to infinity using the Gaussian primes separated by a uniformly bounded length. Some approaches have found the farthest Gaussian prime and the amount of Gaussian primes for a Gaussian Moat of a given length. Nevertheless, such approaches do not provide information regarding the minimum amount of Gaussian primes required to find the desired Gaussian Moat and the number and length of shortest paths of a Gaussian Moat, which become important information in the study of this problem. In this work, we present a computer-based approach to find Gaussian Moats as well as their corresponding minimum amount of required Gaussian primes, shortest paths, and lengths. Our approach is based on the creation of a graph where its nodes correspond to the calculated Gaussian primes. In order to include all Gaussian primes involved in the Gaussian Moat, a backtracking algorithm is implemented. This algorithm allows us to make an exhaustive search of the generated Gaussian primes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Dreyfus, S.E.: An appraisal of some shortest-path algorithms. Oper. Res. 17(3), 395–412 (1969)
Gethner, E., Stark, H.M.: Periodic Gaussian moats. Exp. Math. 6(4), 289–292 (1997)
Gethner, E., Wagon, S., Wick, B.: A stroll through the Gaussian primes. Am. Math. Mon. 105(4), 327–337 (1998)
Ginsberg, M.L.: Dynamic backtracking. J. Artif. Intell. Res. 1, 25–46 (1993)
Guy, R.: Unsolved Problems in Number Theory. Springer, Heidelberg (2004). https://doi.org/10.1007/978-0-387-26677-0
Hernandez, J., Daza, K., Florez, H.: Alpha-beta vs scout algorithms for the Othello game. In: CEUR Workshops Proceedings, vol. 2846 (2019)
Jordan, J., Rabung, J.: A conjecture of Paul Erdo’s concerning Gaussian primes. Math. Comput. 24(109), 221–223 (1970)
Knuth, D.: The Art of Computer Programming 1: Fundamental Algorithms 2: Seminumerical Algorithms 3: Sorting and Searching. Addison-Wesley, Boston (1968)
Loh, P.R.: Stepping to infinity along Gaussian primes. Am. Math. Mon. 114(2), 142–151 (2007)
Oliver, R.J.L., Soundararajan, K.: Unexpected biases in the distribution of consecutive primes. Proc. Nat. Acad. Sci. 113(31), E4446–E4454 (2016)
Prasad, S.: Walks on primes in imaginary quadratic fields. arXiv preprint arXiv:1412.2310 (2014)
Sanchez, D., Florez, H.: Improving game modeling for the quoridor game state using graph databases. In: Rocha, Á., Guarda, T. (eds.) ICITS 2018. AISC, vol. 721, pp. 333–342. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73450-7_32
Tsuchimura, N.: Computational results for Gaussian moat problem. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 88(5), 1267–1273 (2005)
Vardi, I.: Prime percolation. Exp. Math. 7(3), 275–289 (1998)
Velasco, A., Aponte, J.: Automated fine grained traceability links recovery between high level requirements and source code implementations. ParadigmPlus 1(2), 18–41 (2020)
West, P.P., Sittinger, B.D.: A further stroll into the Eisenstein primes. Am. Math. Mon. 124(7), 609–620 (2017)
Acknowledgments
A.C.-A. acknowledges funding from Fundación Universitaria Konrad Lorenz (Project 5INV1). Computational efforts were performed on the High Performance Computing System, operated and supported by Fundación Universitaria Konrad Lorenz.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Florez, H., Cárdenas-Avendaño, A. (2020). A Computer-Based Approach to Study the Gaussian Moat Problem. In: Florez, H., Misra, S. (eds) Applied Informatics. ICAI 2020. Communications in Computer and Information Science, vol 1277. Springer, Cham. https://doi.org/10.1007/978-3-030-61702-8_33
Download citation
DOI: https://doi.org/10.1007/978-3-030-61702-8_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-61701-1
Online ISBN: 978-3-030-61702-8
eBook Packages: Computer ScienceComputer Science (R0)