Year 2020, Volume 2 , Issue 2, Pages 79 - 89 2020-12-30

The Hasse-Minkowski Theorem and Legendre's Theorem for Quadratic Forms in Two and Three Variables

Phuc NGO [1] , Mehmet DİK [2]


Determining the solvability of equations has been an extended and fundamental study in Mathematics. The local-global principle states two objects are equivalent globally if and only if they are equivalent locally at all places. By applying this principle, the Hasse - Minkowski theorem is able to identify the existence of rational solutions of an equation. This paper explores the applications of the Hasse-Minkowski theorem to homogeneous quadratic forms in two and three variables. After providing some of the necessary proofs and definitions, we have been able to introduce some complete computer programs implementing the Hasse-Minkowski theorems and Legendre theorem with some supporting functions like the Eratosthenes sieve.
Hasse-Minkowski, quadratic form, algorithm
  • S. D. Hoehner, The Hasse-Minkowski Theorem in Two and Three Variables(2012). etd.ohiolink.edu/!etd.sendfile?accession=osu1338317481
  • G. A. Jones and J. M. Jones, Elementary Number Theory(Springer, 1998).
  • W. J. LeVeque, Fundamentals of Number Theory(Dover Publications, 1977).
Primary Language en
Subjects Computer Science, Interdisciplinary Application
Journal Section Articles
Authors

Orcid: 0000-0002-9658-4877
Author: Phuc NGO (Primary Author)
Institution: Beloit College
Country: United States


Orcid: 0000-0003-0643-2771
Author: Mehmet DİK
Institution: Beloit College
Country: United States


Dates

Publication Date : December 30, 2020

Bibtex @research article { pims827611, journal = {Proceedings of International Mathematical Sciences}, issn = {2717-6355}, address = {Maltepe University, Istanbul}, publisher = {İbrahim ÇANAK}, year = {2020}, volume = {2}, pages = {79 - 89}, doi = {10.47086/pims.827611}, title = {The Hasse-Minkowski Theorem and Legendre's Theorem for Quadratic Forms in Two and Three Variables}, key = {cite}, author = {Ngo, Phuc and Dik, Mehmet} }