Version 1
: Received: 24 June 2021 / Approved: 28 June 2021 / Online: 28 June 2021 (15:45:56 CEST)
How to cite:
Krishna B, A.; Sai Gopal, M.; Ghosh, S. Special Primes And Some Of Their Properties. Preprints2021, 2021060681. https://doi.org/10.20944/preprints202106.0681.v1
Krishna B, A.; Sai Gopal, M.; Ghosh, S. Special Primes And Some Of Their Properties. Preprints 2021, 2021060681. https://doi.org/10.20944/preprints202106.0681.v1
Krishna B, A.; Sai Gopal, M.; Ghosh, S. Special Primes And Some Of Their Properties. Preprints2021, 2021060681. https://doi.org/10.20944/preprints202106.0681.v1
APA Style
Krishna B, A., Sai Gopal, M., & Ghosh, S. (2021). Special Primes And Some Of Their Properties. Preprints. https://doi.org/10.20944/preprints202106.0681.v1
Chicago/Turabian Style
Krishna B, A., Mantha Sai Gopal and Sourangshu Ghosh. 2021 "Special Primes And Some Of Their Properties" Preprints. https://doi.org/10.20944/preprints202106.0681.v1
Abstract
In this paper, we present the definition, some properties, and solve a problem on special primes. These properties help in providing us with a better understanding of the problem posed related to special primes on the open problem garden website. The problem involves finding all the primes q, given a prime p such that q≡1(mod p) and 2^((q−1)/p)≡1(mod q). We prove that a prime number q is a special prime of p if and only if the order of 2 in U(q) divides q−1p. Also, we prove that a prime number q is not a special prime for any prime number if 2 is a generator of the group U(q) and that there exist infinitely many special primes for any given prime number.
Keywords
Special Primes, Cubic Reciprocity, Primitive Roots, Artin’s conjecture
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
