Over millennia, people have considered prime numbers as unpredictable and hard to list. This study confirms that all positive integers without factors of 2, 3, 5 and 7 can be regarded as the offspring of 48 roots in the interval [11, 220]. These roots are to generate a Factor-Pair Table composed of 48 columns, each of which forms a basic binary matrix to indicate the locations of composite numbers. For a given interval, these composites are removed via mapping effect. After primality checking of the rest of the numbers, we identify all primes within the interval. We then form a Formula of Primes for prime prediction. Moreover, the Periodic Table of Primes (PTP) [1] is reconstructed based on the proposed algorithm to illustrate the distribution of primes and composites.
References
[1]
Li, H., Fang, S. and Kuo, W. (2024) The Periodic Table of Primes. AdvancesinPureMathematics, 14, 394-419. https://doi.org/10.4236/apm.2024.145023
[2]
Bach, E. and Shallit, J. (1996) Algorithmic Number Theory. MIT Press.
[3]
O’Neill, M.E. (2009) The Genuine Sieve of Eratosthenes. JournalofFunctionalProgramming, 19, 95-106. https://doi.org/10.1017/s0956796808007004
Atkin, A.O.L. and Bernstein, D.J. (2003) Prime Sieves Using Binary Quadratic Forms. MathematicsofComputation, 73, 1023-1030. https://doi.org/10.1090/s0025-5718-03-01501-1
[6]
Pritchard, P. (1983) Fast Compact Prime Number Sieves (among Others). JournalofAlgorithms, 4, 332-344. https://doi.org/10.1016/0196-6774(83)90014-7
[7]
Helfgott, H.A. (2019) An Improved Sieve of Eratosthenes. MathematicsofComputation, 89, 333-350. https://doi.org/10.1090/mcom/3438
[8]
Pritchard, P. (1982) Explaining the Wheel Sieve. ActaInformatica, 17, 477-485. https://doi.org/10.1007/bf00264164
[9]
Agrawal, M., Kayal, N. and Saxena, N. (2004) PRIMES Is in P. AnnalsofMathematics, 160, 781-793. https://doi.org/10.4007/annals.2004.160.781
[10]
Miller, G.L. (1975) Riemann’s Hypothesis and Tests for Primality. Proceedings of Seventh Annual ACM Symposium on Theory of Computing, Albuquerque, 5-7 May 1975, 234-239. https://doi.org/10.1145/800116.803773
