Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 13, 30 November 2023
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Factoring primes and sums of two squares
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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
References
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.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-189-6
- ISBN (Online)
- 978-1-83558-190-2
- Published Date
- 30 November 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/13/20240744
- Copyright
- 30 November 2023
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited