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

Year 2021, Volume: 3 Issue: 2, 98 - 108, 30.12.2021

Phuc Ngo Mehmet Dik

Abstract

Our paper was hugely inspired by Dr. Hohner’s master thesis, “The Hasse-Minkowski Theorem in Two and Three Variables.” More than half the length of our paper is our original programming implementation of various theorems, like the Hasse-­Minkowski theorem and Legendre’s theorem, and many supporting concepts, along with the algorithm analysis. We also shorten many proofs from Dr. Hohner’s paper by either providing an alternative shorter version or summarizing them. We credit him in section 1 on the binary and ternary quadratic form and the bibliography. However, the location of the credit section 1 was supposed to be before section 1, and this is a formatting mistake. Even though we made an effort to credit Dr. Hohner’s work, it could still be insufficient. We think it would be best to retract the paper for those listed reasons.

Keywords

Hasse-Minkowksi, quadratic form, algorithm

References

• [1] S. D. Hoehner, The Hasse-Minkowski Theorem in Two and Three Variables (2012). [2] G. A. Jones and J. M. Jones, Elementary Number Theory (Springer, 1998). [3] W. J. LeVeque, Fundamentals of Number Theory (Dover Publications, 1977).