Traditionally, primes have been viewed as weeds appearing irregularly among the natural numbers, making their locations difficult to predict. We believe composite numbers, the complement of primes, exhibit symmetric patterns in their distribution. This study presents a new perspective: after systematically eliminating integers divisible by small primes such as 2, 3, 5, and 7, the remaining integers can be uniquely represented through a framework of roots and kins. Furthermore, all remaining composite numbers exhibit cyclic and mirror effects, enabling the construction of an extended Cyclic Table of Composites. Using this table, we derive a formula that identifies the locations of primes within any interval. Additional formulas for fast factorization and for locating twin primes, mirror primes, and prime tuples are also obtained. These results suggest that although primes themselves appear structureless, the composites surrounding them obey rich internal symmetries that allow efficient detection of “where primes hide”.
References
[1]
Dickson, L.E. (2005) History of the Theory of Numbers, Volume II: Diophantine Analysis. Dover Publications.
[2]
Zagier, D. (1977) The First 50 Million Prime Numbers. The Mathematical Intelligencer, 1, 7-19. https://doi.org/10.1007/bf03039306
[3]
Stein, M.L., Ulam, S.M. and Wells, M.B. (1964) A Visual Display of Some Properties of the Distribution of Primes. TheAmericanMathematicalMonthly, 71, 516-520. https://doi.org/10.2307/2312588
[4]
Ribenboim, P. (1996) The New Book of Prime Number Records. Springer. https://doi.org/10.1007/978-1-4612-0759-7
[5]
Crandall, R. and Pomerance, C. (2005) Prime Numbers: A Computational Perspective. Springer.
[6]
Aiyar, V.R. (1934) Sundaram’s Sieve for Prime Numbers. The Mathematics Student, 2, 73.
[7]
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
[8]
Pritchard, P. (1982) Explaining the Wheel Sieve. Acta Informatica, 17, 477-485. https://doi.org/10.1007/bf00264164
[9]
Kumanduri, R. and Romero, C. (1998) Number Theory with Computer Applications. Prentice Hall.
[10]
Matiyasevich, Y.V. (1999) Formulas for Prime Numbers. In: Tabachnikov, S., Ed., Kvant Selecta: Algebra and Analysis (Vol. 2), American Mathematical Society, 13-24.
[11]
Zagier, D. (1997) Newman’s Short Proof of the Prime Number Theorem. TheAmericanMathematicalMonthly, 104, 705-708. https://doi.org/10.1080/00029890.1997.11990704
[12]
Li, H., Fang, S., Lin, B.M.T. and Kuo, W. (2023) Unifying Colors by Primes. Light: Science&Applications, 12, Article No. 32. https://doi.org/10.1038/s41377-023-01073-x
[13]
Li, H., Huang, Y., Fang, S. and Kuo, W. (2021) A Prime-Logarithmic Method for Optimal Reliability Design. IEEETransactionsonReliability, 70, 146-162. https://doi.org/10.1109/tr.2020.3020597
[14]
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
[15]
Li, H., Fang, S., Kuo, W. and Lin, N. (2025) Listing Prime Numbers Periodically. AdvancesinPureMathematics, 15, 247-268. https://doi.org/10.4236/apm.2025.154012
[16]
Li, H., Fang, S., Kuo, W. and Lin, N. (2025) Kernel Factor Pairs for Semiprime Factorization. AdvancesinPureMathematics, 15, 629-642. https://doi.org/10.4236/apm.2025.159032
