## A Logical Proof of the Polignac's Conjecture Based on Partitions of an Even Number of a New Formulation

Published: 2024-03-16

Page: 1-11

Daniel Sankei *

Department of Mathematics, Meru University of Science and Technology, P.O. Box 972, Meru, Kenya.

Loyford Njagi

Department of Mathematics, Meru University of Science and Technology, P.O. Box 972, Meru, Kenya.

Josephine Mutembei

Department of Mathematics, Meru University of Science and Technology, P.O. Box 972, Meru, Kenya.

*Author to whom correspondence should be addressed.

### Abstract

Polignac's Conjecture, proposed by Alphonse de Polignac in the 19th century, is a captivating hypothesis that extends the notion of twin primes to a broader context. It posits that for any even positive integer 2k, there exist infinitely many pairs of consecutive prime numbers whose difference is 2k. This conjecture is a natural generalization of the Twin Prime Conjecture, which focuses solely on pairs of primes differing by two. The conjecture has significant implications for our understanding of the distribution of prime numbers and the nature of their gaps and its exploration serves as a testament to the enduring fascination and mystery surrounding prime numbers and their properties. However, despite extensive efforts by mathematicians over the years, Polignac's Conjecture remains unproven, standing as one of the many unsolved problems in number theory.   This study utilizes a set of all odd partitions generated from an even number of a new formulation, and we show that from this set of all pairs of odd numbers, there exist proper subsets containing infinitely many pairs of prime numbers whose differences is a fixed even gap. Finally, using these results and the facts that the difference of any two prime numbers is even and there exist infinitely many prime numbers, a logical proof of the Polignac's Conjecture is provided.

Keywords: Polignac's conjecture, even numbers, odd numbers, prime numbers

#### How to Cite

Sankei, Daniel, Loyford Njagi, and Josephine Mutembei. 2024. “A Logical Proof of the Polignac’s Conjecture Based on Partitions of an Even Number of a New Formulation”. Asian Research Journal of Mathematics 20 (3):1-11. https://doi.org/10.9734/arjom/2024/v20i3787.

### References

Mothebe MF, Kagiso DN, Modise BT. A proof of cases of de Polignac's conjecture; 2019.

arXiv preprint arXiv:1912.09290.

Orús-Lacort M, Orús R, Jouis C. Analyzing twin primes, Goldbach’s strong conjecture and Polignac’s conjecture; 2023. Available:https://doi.org/10.20944/preprints202311.1660.v1

Ting JY. Solving Polignac’s and twin prime conjectures using information-complexity conservation; 2017.

Okouma PM, Hawing G. Novel aspects of the global regularity of primes; 2022.

arXiv preprint arXiv:2310.03973.

Neale V. Closing the gap: The quest to understand prime numbers. Oxford University Press; 2017

Debnath P. Open problems in number theory. In Advances in Number Theory and Applied Analysis. 2023;1-11.

Sankei D, Njagi L, Mutembei J. Partitioning an even number of the new formulation into all pairs of odd numbers. Journal of Mathematical Problems, Equations and Statistics. 2023;4(2):35-37. Available:https://doi. org/10.22271/math. 2023. v4. i2a, 104

Pintz J. Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz. 2016;367-384.

Sankei D, Njagi L, Mutembei J. A new formulation of a set of even numbers. European Journal of Mathematics and Statistics. 2023;4(4):93-97.

Hirschhorn MD. There are infinitely many prime numbers. Australian Mathematical Society Gazette. 2002;29(2):103-103.

Goldston DA. Are there infinitely many primes? 2007.

arXiv preprint arXiv:0710.2123.

Verification of a proof that the difference of two odd integers is not odd. (n.d.). Mathematics Stack Exchange. Available:https://math.stackexchange.com/questions/1058542/verification-of-a-proof-that-the-difference-of-two-odd-integers-is-not-odd

Taylor S. Combination. Corporate Finance Institute; 2023. Available:https://corporatefinanceinstitute.com/resources/data-science/combination/

Daniel S, Njagi L, Mutembei J. A numerical verification of the strong Goldbach conjecture up to 9× 10^ 18. GPH-International Journal of Mathematics. 2023;6(11):28-37.

Available:https://en.wikipedia.org/wiki/Twin_prime

Mothebe MF, Kagiso DN, Modise BT. A proof of cases of de Polignac's conjecture; 2019.

arXiv preprint arXiv:1912.09290.