Download pdf
Factoring primes and sums of two squares
- 1 The University of Manchester
* Author to whom correspondence should be addressed.
Abstract
Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .
Keywords
Primes, Sums of Squares, Algebraic Number Theory, Mathematics
[1]. Bhaskar, J. (2008). Sum of two squares. https://www.math.uchicago.edu/~may/VIGRE/ VIGRE2008/REUPapers/Bhaskar.pdf.
[2]. Homeworkhelp. Com and Inc. Factoring and Primes (High School Math).
[3]. Stewart, I. and Tall, D. (2001). Algebraic Number Theory and Fermat's Last Theorem: Third Edition. A K Peters/CRC Press. ISBN-10: 1568811195. ISBN-13: 978-1568811192.
[4]. Edwards, H. M. (1977). Fermat's last theorem: A genetic introduction to algebraic number theory. In: Graduate Texts in Mathematics. Springer New York, NY.
[5]. Stein, M. R. and Dennis, R. K. (1989). Algebraic K-Theory and Algebraic Number Theory: Comtemporar Math., 83. American Mathematical Society, Providence.
[6]. Rosen, K. H. (2000). Elementary number theory and its applications. Addison Wesley. ISBN-10: 0201870738. ISBN-13: 978-0201870732.
[7]. Zagier, D. (1990). A One-Sentence Proof That Every Prime p Is a Sum of Two Squares. In: American Mathematical Monthly 97(2), 144.
[8]. Honsberger, R. (1970). Writing a Number as a Sum of Two Squares. In Ingenuity In Mathematics (Anneli Lax New Mathematical Library, pp. 61-66). Mathematical Association of America. doi:10.5948/UPO9780883859384.012.
[9]. Vladimirovich, D. V. and Genadievna, S. A. (2017). A generalization of fermat’s theorem on sum of two squares. Austrian Journal of Technical and Natural Sciences.
[10]. Ore, O. (1988). Number theory and its history (Dover Books on Mathematics). Dover Publications. ISBN-10: 0486656209. ISBN-13: 978-0486656205.
Cite this article
Jiang,Z. (2023). Factoring primes and sums of two squares. Theoretical and Natural Science,13,18-22.
Data availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
Disclaimer/Publisher's Note
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
About volume
Volume title: Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license. Authors who
publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons
Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this
series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published
version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial
publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and
during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See
Open access policy for details).