Research ArticleOpen AccessGoogle Scholar indexed
Group-Theoretic Remarks on Goldbach’s Conjecture
Department of Mathematics, Shenyang University of Technology, Shenyang, China
College of Teacher Education, Harbin University, Harbin, China
- 1 Department of Mathematics, Shenyang University of Technology, Shenyang, China
- 2 College of Teacher Education, Harbin University, Harbin, China
Advances in Pure Mathematics·Volume 12 (2022)·Pages 624–637·Published 2 November 2022·DOI10.4236/apm.2022.1211048
Copy link · social · email
Abstract
The famous strongly binary Goldbach’s conjecture asserts that every even number 2 n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees 2 n and 2 n − 1 to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for p +1 and p − 1 whenever p ≥ 11 is a prime number.
KeywordsAlternating GroupElement Order Prime GraphGoldbach’s ConjectureCentralizer
- Goldbach’s Conjecture. https://en.wikipedia.org/wiki/Goldbach
- Li, K. (2022) A New Method to Study Goldbach Conjecture. Applied Mathematics, 13, 68-76. https://doi.org/10.4236/am.2022.131006
- Isaacs, I.M. (1994) Algebra: A Graduate Textbook. Brooks/Cole, Pacific Grove.
- Lucido, M.S. (1999) Prime Graph Components of Finite Almost Simple Groups. Rendiconti del Seminario Matematico della Università di Padova, 102, 1-22. (Addendum, 2002, 107, 89-90).
- Williams, J.S. (1981) Prime Graph Components of Finite Groups. Journal of Algebra, 69, 487-513. https://doi.org/10.1016/0021-8693(81)90218-0
- (2012) GAP-Groups Algorithms, and Programming, Version 4.5. https://www.gap-system.org
- Gorshkov, I.B. (2013) Recognizability by Spectrum of Alternating Groups. Algebra and Logic, 52, 41-46. https://doi.org/10.1007/s10469-013-9217-x
- Isaacs, I.M. (1976) Character Theory of Finite Groups. Academic Press, New York.
- Kurzweil, H. and Stellmacher, B (2004) The Theory of Finite Groups: An Introduction. Springer-Verlag, New York. https://doi.org/10.1007/b97433
- Lewis, M. (2008) An Overview of Graphs Associated with Character Degrees and Conjugacy Class Sizes in Finite Groups. Rocky Mountain Journal of Mathematics, 38, 175-211. https://doi.org/10.1216/RMJ-2008-38-1-175
- Vasil’e, A.V. and Vdovin, E.P. (2005) An Adjacency Criterion for the Prime Graph of a Finite Simple Group. Algebra and Logic, 44, 381-406. https://doi.org/10.1007/s10469-005-0037-5
- Miller, W. (1987) The Maximum Order of an Element of a Finite Symmetric Group. The American Mathematical Monthly, 94, 497-506. https://doi.org/10.1080/00029890.1987.12000673
- Cohen, H. (2007) Number Theory Volume II: Analytic and Modern Tools. Springer-Verlag, New York.
- Rosser, J.S. and Schoenfeld, L. (1962) Approximate Formulas for Some Functions of Prime Numbers. Illinois Journal of Mathematics, 6, 64-94. https://doi.org/10.1215/ijm/1255631807
- Higman, G. (1957) Finite Groups in Which Every Element Has Prime Power Order. Journal of the London Mathematical Society, s1-32, 321-334. https://doi.org/10.1112/jlms/s1-32.3.321
- Bi, J.X. (2001) Characterization of Alternating Groups by Orders of Normalizers of Sylow Subgroups. Algebra Colloquium, 8, 13-15.