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

Phuc Ngo; Mehmet Dik

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

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

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).

Details

Primary Language

English

Subjects

Software Engineering (Other)

Journal Section

Research Article

Authors

Phuc Ngo ^* This is me
0000-0002-9658-4877
United States

Mehmet Dik This is me
0000-0003-0643-2771
United States

Publication Date

December 30, 2021

Submission Date

August 8, 2020

Acceptance Date

December 30, 2020

Published in Issue

Year 2021 Volume: 3 Number: 2

IZ

https://izlik.org/JA57EN33DP

Cite

RIS / Bibtex

APA

Ngo, P., & Dik, M. (2021). The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables. Proceedings of International Mathematical Sciences, 3(2), 98-108. https://izlik.org/JA57EN33DP

AMA

1.Ngo P, Dik M. The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables. PIMS. 2021;3(2):98-108. https://izlik.org/JA57EN33DP

Chicago

Ngo, Phuc, and Mehmet Dik. 2021. “The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables”. Proceedings of International Mathematical Sciences 3 (2): 98-108. https://izlik.org/JA57EN33DP.

EndNote

Ngo P, Dik M (December 1, 2021) The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables. Proceedings of International Mathematical Sciences 3 2 98–108.

IEEE

[1]P. Ngo and M. Dik, “The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables”, PIMS, vol. 3, no. 2, pp. 98–108, Dec. 2021, [Online]. Available: https://izlik.org/JA57EN33DP

ISNAD

Ngo, Phuc - Dik, Mehmet. “The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables”. Proceedings of International Mathematical Sciences 3/2 (December 1, 2021): 98-108. https://izlik.org/JA57EN33DP.

JAMA

1.Ngo P, Dik M. The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables. PIMS. 2021;3:98–108.

MLA

Ngo, Phuc, and Mehmet Dik. “The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables”. Proceedings of International Mathematical Sciences, vol. 3, no. 2, Dec. 2021, pp. 98-108, https://izlik.org/JA57EN33DP.

Vancouver

1.Phuc Ngo, Mehmet Dik. The Hasse-Minkowski Theorem and Legendre’s Theorem for Quadratic Forms in Two and Three Variables. PIMS [Internet]. 2021 Dec. 1;3(2):98-108. Available from: https://izlik.org/JA57EN33DP