How Prime Numbers Are Interconnected and Built with Two Equations: Addition and Subtraction Rules the Function (6µ)
- 1 Centre of Pedagogical Innovation (Capsule), Sorbonne University, Paris, France
Abstract
Are all prime numbers linked by four simple functions? Can we predict when a prime will appear in a sequence of primes? If we classify primes into two groups, Group 1 for all primes that appear before <i>ζ</i> (such that <img src="https://html.scirp.org//file/5302399-rId13.svg?20240416020453" >, for instance 5, <img src="https://html.scirp.org//file/5302399-rId15.svg?20240416020453" >), an even number divisible by 3 and 2, and Group 2 for all primes that are after <i>ζ</i> (such that <img src="https://html.scirp.org//file/5302399-rId17.svg?20240416020453" >, for instance 7), then we find a simple function: for each prime in each group, <img src="https://html.scirp.org//file/5302399-rId19.svg?20240416020453" >, where <i>n</i> is any natural number. If we start a sequence of primes with 5 for Group 1 and 7 for Group 2, we can attribute a <i>μ</i> value for each prime. The <i>μ</i> value can be attributed to every prime greater than 7. Thus <img src="https://html.scirp.org//file/5302399-rId21.svg?20240416020453" > for Group 1, and <img src="https://html.scirp.org//file/5302399-rId23.svg?20240416020453" >. Using this formula, all the primes appear for <img src="https://html.scirp.org//file/5302399-rId25.svg?20240416020453" >, where <i>μ</i> is any natural number.
- Thomas, I. (1991) Greek Mathematical Works. Loeb Classical Library, London, Volume I.
- Williamson, J. (1788) The Elements of Euclid. With Dissertations ... by James Williamson. Clarendon Press, Oxford, Volume IX.
- Stillwell, J. (2003) The Euclidean Algorithm. In: Elements of Number Theory . Undergraduate Texts in Mathematics. Springer, New York. https://doi.org/10.1007/978-0-387-21735-2_2
- Erdős, P. and Surányi, J. (2003) Properties of Prime Numbers. In: Topics in the Theory of Numbers . Undergraduate Texts in Mathematics. Springer, New York, 157-180. https://doi.org/10.1007/978-1-4613-0015-1_5
- Maynard, J. (2019) The Twin Prime Conjecture. Japanese Journal of Mathematics , 14, 175-206. https://doi.org/10.1007/s11537-019-1837-z
- Zhang, Y.T. (2014) Bounded Gaps between Primes. Annals of Mathematics , 179, 1121-1174. https://doi.org/10.4007/annals.2014.179.3.7
- Liu, F.S. (2020) There Are Infinitely Many Fibonacci Primes. Global Journal of Science Frontier Research , 20, 1-17. https://doi.org/10.34257/GJSFRFVOL20IS5PG1
- Panzone, P.A. (2012) On the Generating Functions of Mersenne and Fermat Primes. Collectanea Mathematica , 63, 59-69. https://doi.org/10.1007/s13348-010-0019-3
- Abdel-Mageed, M., Salim, A., Osamy, W. and Khedr, A.M. (2021) A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories. Advances in Mathematical Physics , 2021, Article ID 9956518. https://doi.org/10.1155/2021/9956518
- Green, B. (2000) The Prime Number Theorem. Mathematical Institute, Oxford.
- Green, B. and Tao, T. (2008) The Primes Contain Arbitrarily Long Arithmetic Progressions. Annals of Mathematics , 167, 481-547. https://doi.org/10.4007/annals.2008.167.481
- Booker, A.R. (2016) A Variant of the Euclid-Mullin Sequence Containing Every Prime. arXiv preprint, arXiv:1605.08929.