Abhishek Singh,
Dhyey Shingala,
- Research Scholar, Department of Mathematics and Physics, Texas Tech University, Lubbock, USA
- Student, Department of Computer Science, Texas Tech University, Lubbock, USA
Abstract
In this paper, we present a theorem and an efficient algorithm for computing all prime numbers up to a given integer X. Our theorem establishes a relationship between the primes up to X and those up to (X+1)2, providing a theoretical foundation for the algorithm. The proposed method improves upon the traditional Sieve of Eratosthenes by dynamically eliminating multiples of primes using only the remaining numbers in the set, thereby reducing redundant computations and enhancing performance Furthermore, we introduce a novel set-theoretic framework for analysing Prime Gaps. Specifically, we define the Gap Set for a prime p as the sequence of gaps between multiples of p in the remaining set of numbers after removing multiples of all previous primes. Remarkably, these gap sets exhibit a repeating pattern, which we rigorously analyse to uncover a Recursive Structure in the distribution of primes. This observation not only deepens our understanding of prime numbers but also reveals a striking relationship between consecutive primes: if p and q are consecutive primes, the size of the gap set for p is given by Size(p)=(q−1)×Size(q).While this relationship lacks a formal proof, extensive empirical evidence supports its validity across a wide range of primes. Our work bridges theoretical and computational approaches, providing both a theorem and a practical algorithm for Prime Number Generation. By combining insights from the recursive structure of gap sets and the observed relationship between consecutive primes, we offer a fresh perspective on the distribution of primes and a potential pathway for predicting their positions—a problem that has remained unsolved for centuries. The results demonstrate significant improvements in efficiency and contribute to the ongoing quest to unravel the mysteries of prime numbers
Keywords: Gap set, prime number generation, recursive structure, prime gaps
[This article belongs to Research & Reviews: Discrete Mathematical Structures ]
Abhishek Singh, Dhyey Shingala. Dynamics of Prime Gap Sets: An Algorithmic and Set-Theoretic Approach. Research & Reviews: Discrete Mathematical Structures. 2025; 12(03):20-25.
Abhishek Singh, Dhyey Shingala. Dynamics of Prime Gap Sets: An Algorithmic and Set-Theoretic Approach. Research & Reviews: Discrete Mathematical Structures. 2025; 12(03):20-25. Available from: https://journals.stmjournals.com/rrdms/article=2025/view=230487
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Research & Reviews: Discrete Mathematical Structures
| Volume | 12 |
| Issue | 03 |
| Received | 24/10/2025 |
| Accepted | 28/10/2025 |
| Published | 30/10/2025 |
| Publication Time | 6 Days |
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