Statistical Analysis and Distribution of Fermat Pseudoprimes Within the Given Interval

Bhawana Fulara; Arvind Bhatt; Deepak Kumar Sharma; Shubham Agarwal; Geeta Mathpal; Rajesh Mathpal

doi:10.52756/ijerr.2024.v44spl.010

Authors

Bhawana Fulara Department of Mathematics, Uttarakhand Open University, Haldwani-263139, Nainital, India https://orcid.org/0009-0008-0153-2128
Arvind Bhatt Department of Mathematics, Uttarakhand Open University, Haldwani-263139, Nainital, India https://orcid.org/0000-0001-5010-1510
Deepak Kumar Sharma Department of Mathematics, Uttarakhand Open University, Haldwani-263139, Nainital, India https://orcid.org/0009-0005-9819-2753
Shubham Agarwal Department of Mathematics, New Delhi Institute of Management, New Delhi-110062, Delhi, India https://orcid.org/0000-0001-9205-9904
Geeta Mathpal Department of Mathematics, Uttarakhand Open University, Haldwani-263139, Nainital, India https://orcid.org/0009-0004-6621-907X
Rajesh Mathpal Department of Physics, Uttarakhand Open University, Haldwani-263139, Nainital, India https://orcid.org/0009-0003-4361-6630

DOI:

https://doi.org/10.52756/ijerr.2024.v44spl.010

Keywords:

Algorithm, analysis, distribution, interval, pseudoprimes

Abstract

Prime numbers are natural numbers that can only be divided by one and the original number. There is more than one of them. Error-correcting codes used in telecommunications are generated using prime numbers. They guarantee automatic message correction both during transmission and reception. Algorithms used in public-key cryptography are built upon primes. They're also employed in the production of pseudorandom numbers. Mathematicians and many other scientific and technological communities have always been fascinated by prime numbers. Additionally, computer engineers can use it to tackle a wide range of real-world problems. The analysis of prime numbers is very important for finding their applications in different fields. The statistical analysis of pseudoprimes within a given interval is carried out in the presented paper and an algorithm of Python program to find the distribution of pseudoprimes has also been generated, which is used to find their distribution with different bases within the given intervals. The data analysis process made use of graphical depiction. The discovery will surely open up new avenues for future number theory study and applications outside of mathematics.

References

Agarwal, A., Agarwal, S., & Singh, B. K. (2021). Analysis of Fibonacci primes & their application in cryptography. Stochastic Modeling and Applications (SMA), 25(2), 73–82. https://doi.org/10.5281/zenodo.13969700

Agarwal, A., Agarwal, S., & Singh, B. K. (2023). Analysis of primes and developing correlation model between them. Journal of the Maharaja Sayajirao University of Baroda, 57(1), 78–82. https://doi.org/10.5281/zenodo.13969833

Agarwal, S., Sharma, D., & Uniyal, A. S. (2021). Formulation & distribution of super primes. Global and Stochastic Analysis (GSA), 8(2), 155–166. https://doi.org/10.5281/zenodo.13969867

Agarwal, S., & Uniyal, A. S. (2015). Multiprimes distribution within a given norms. International Journal of Applied Mathematical Sciences (JAMS), 8(2), 126–132. https://doi.org/10.5281/zenodo.13970797

Agarwal, S., & Uniyal, A. S. (2018). Algorithms for number theoretic functions & special numbers. International Journal of Research in Engineering, Science and Management (IJRESM), 1(12), 112–116. https://doi.org/10.5281/zenodo.13969746

Erd?s, P. (1956). On pseudoprimes and Carmichael numbers. Publ. math. Debrecen, 4, 201–206. DOI: https://doi.org/10.5486/PMD.1956.4.3-4.16

Gradini, E. (2010). Fermat test and the existence of pseudoprimes. Visipena Journal, 1(1), 37–44. https://doi.org/10.46244/visipena.v1i1.21 DOI: https://doi.org/10.46244/visipena.v1i1.21

Hamahata, Y., & Kokubun, Y. (2007). Cipolla pseudoprimes. Journal of Integer Sequences, 10, 1-6. http://eudml.org/doc/54790

He, T. X., Shiue, P. J. S., & Chang, Y. (2022). Computation of Fermat’s pseudoprimes (Dedicated to the Memory of Professor Leetsch C. Hsu). Journal of Discrete Mathematical Sciences and Cryptography, 25(2), 335–352. https://doi.org/10.1080/09720529.2019.1662580 DOI: https://doi.org/10.1080/09720529.2019.1662580

Jaeschke, G. (1993). On strong pseudoprimes to several bases. Mathematics of Computation, 61, 915–926. https://doi.org/10.1090/S0025-5718-1993-1192971-8 DOI: https://doi.org/10.1090/S0025-5718-1993-1192971-8

Jiang, Y., & Deng, Y. (2014). Strong pseudoprimes to the first eight prime bases. Mathematics of Computation, 83, 2915–2924. https://doi.org/10.1090/S0025-5718-2014-02830-5 DOI: https://doi.org/10.1090/S0025-5718-2014-02830-5

K?ížek, M., Luca, F., & Somer, L. (2002). Fermat’s little theorem, pseudoprimes, and super pseudoprimes. Springer, In 17 Lectures on Fermat Numbers, pp. 317–338. https://doi.org/10.1007/978-0-387-21850-2_12 DOI: https://doi.org/10.1007/978-0-387-21850-2_12

Li, S. (1996). On the distribution of even pseudoprimes.

Ordowski, T. (2021). Density of Fermat weak pseudoprimes k to a base d, where d | k and 1 < d < k. Retrieved from http://list.seqfan.eu/pipermail/seqfan/2021-January/073021.html

Parhi, K., & Kumari, P. (2018). Properties of strong pseudoprimes on base b. International Journal of Creative Research Thoughts, 6(2), 1306–1310. Retrieved from http://www.ijcrt.org/papers/IJCRT1813261.pdf

Pomerance, C. (1981). On the distribution of pseudoprimes. Mathematics of Computation, 37, 587–593. https://doi.org/10.1090/S0025-5718-1981-0628717-0 DOI: https://doi.org/10.1090/S0025-5718-1981-0628717-0

Pomerance, C., & Samuel, S. W. (2023). Some thoughts on pseudoprimes. CERIAS Center at Purdue University, 1–11.

Pushpa, A. M., & Subramanian, S. (2021). Study of prime, pseudoprime and applications of pseudoprime. Turkish Journal of Computer and Mathematics Education, 12(9), 934–939. Retrieved from https://turcomat.org/index.php/turkbilmat/article/view/3333

Ribenboim, P. (1996). How Are the Prime Numbers Distributed? In: The New Book of Prime Number Records. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0759-7_5 DOI: https://doi.org/10.1007/978-1-4612-0759-7

Rotkiewicz, A. (1967). On the pseudoprimes of the form ax + b. Mathematical Proceedings of the Cambridge Philosophical Society, 63(2), 389–392. https://doi.org/10.1017/S030500410004130X DOI: https://doi.org/10.1017/S030500410004130X

Sharma, D., Agarwal, S., & Uniyal, A. S. (2021). Distribution of multi-reverse primes within the given interval & their application in asymmetric cryptographic algorithm. International Journal of Applied Engineering and Technology, 3(1), 29–33. https://doi.org/10.5281/zenodo.13969790

Sharma, D., Agarwal, S., & Uniyal, A. S. (2022a). Neoteric relationship between various primes and their analysis. Stochastic Modeling and Applications, 26(1), 155–162. https://doi.org/10.5281/zenodo.13969806

Sharma, D., Agarwal, S., & Uniyal, A. S. (2022b). Linear regression model for various primes. Journal of the Maharaja Sayajirao University of Baroda, 56(1), 16–23. https://doi.org/10.5281/zenodo.13970738

Somer, L. (1987). On Fermat d-pseudoprimes. In J. M. de Koninck & C. Levesque (Eds.), Théorie des nombres / Number theory. De Gruyter, pp. 841–860. https://doi.org/10.1515/9783110852790.841 DOI: https://doi.org/10.1515/9783110852790.841

Wagstaff, S. S., Jr. (2024). Pseudoprimes and Fermat numbers. Integers: 24, 1–10. https://doi.org/10.5281/zenodo.10680422

Zhang, Z. (2007). Two kinds of strong pseudoprimes up to 10. Mathematics of Computation, 76, 2095–2108. https://doi.org/10.1090/S0025-5718-07-01977-1 DOI: https://doi.org/10.1090/S0025-5718-07-01977-1