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Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers
Institut de Química Computacional i Catàlisi, Universitat de Girona, Campus de Montilivi, Girona (Catalonia), Spain
- 1 Institut de Química Computacional i Catàlisi, Universitat de Girona, Campus de Montilivi, Girona (Catalonia), Spain
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Abstract
A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M N = 2 N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0; M N ]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.
KeywordsMersenne NumbersRecursive Generation of Natural NumbersMersenne Natural Number IntervalsCounting the Number of Prime Numbers in Mersenne Natural IntervalsCorrelation between Prime Number Set Cardinalities and Mersenne NumbersExtended Twin Prime Number Conjecture
- Carbó-Dorca, R. (2020) Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling, 3, 120-129. https://doi.org/10.33187/jmsm.776898
- Weisstein, E.W. (n.d.) Mersenne Number. MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/MersenneNumber.html
- Derbyshire, J. (2004) Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. A Plume Book, Penguin Group.
- https://www.mersenne.org/primes/
- Goldston, D.A., Pintz, J. and Yildirin, C.Y. (2005) Primes in Tuples I. arXiv:math\0508185v1 [math.NT] 10 Aug.
- Maynard, J. (2019) Gaps between Primes. Proceedings of the International Congress of Mathematicians, 345-361. https://doi.org/10.1142/9789813272880_0057
- Balasubramaniam, K. and Carbó-Dorca, R. (2022) A Conjecture on Extended Twin Primes. Research Gate. https://doi.org/10.13140/RG.2.2.22432.25600
- Carbó-Dorca, R. (2022) Shadows’ Hypercube, Vector Spaces, and Non-Linear Optimization of QSPR Procedures. Journal of Mathematical Chemistry, 60, 283-310. https://doi.org/10.1007/s10910-021-01301-y
- Carbó-Dorca, R. (2019) Universal Transformation and Non-Linear Connection between Experimental and Calculated Property Vectors in QSPR. Journal of Mathematical Chemistry, 57, 1075-1087. https://doi.org/10.1007/s10910-019-01009-0