Research ArticleOpen AccessGoogle Scholar indexed
Detecting a Regularity in the Generation and Utilization of Primes in the Multiplicative Number Theory
Department of Mathematics and Statistics, York University, Toronto, Canada
- 1 Department of Mathematics and Statistics, York University, Toronto, Canada
Copy link · social · email
Abstract
If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2 p . In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.
KeywordsGoldbach’s ConjectureSymmetric Prime CousinsSystemic Approach in Number TheoryParallel System Covering Integers with PrimesEuler’s Formula for the Product of Reciprocals of PrimesFormula for the Exact Number of Primes Less than or Equal to an Arbitrary Bound
- Guiasu, S. (2009) Probabilistic Models in Operations Research. Nova Science Publishers, New York.
- Hardy, G.H. and Wright, E.M. (1962) Introduction to the Theory of Numbers. 4th Edition, Clarendon Press, Oxford.
- Guy, R.K. (1981) Unsolved Problems in Number Theory. Springer-Verlag, New York-Heidelberg-Berlin.
- Guiasu, S. (1995) Is There Any Regularity in the Distribution of Prime Numbers at the Beginning of the Sequence of Positive Integers? Mathematics Magazine, 68, 110-121. https://doi.org/10.1080/0025570x.1995.11996292
- Ribenboim, P. (2004) The Little Book of Bigger Primes. 2nd Edition, Springer-Verlag, New York.
- Legendre, A.-M. (1830) Théorie des Nombres. 3rd Edition, Firmin Didot Freres, Paris.
- Lehmer, D.H. (1959) On Exact Number of Primes Less Than a Given Bound. Illinois Journal of Mathematics, 3, 381-388. https://doi.org/10.1215/ijm/1255455259
- Wolfram, S. (1991) Mathematica. A System of Doing Mathematics by Computer. 2nd Edition, Addison Wesley Publishing Company, Redwood City, California.