Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Special Primes And Some Of Their Properties

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. 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. Preprints 2021, 2021060681. 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 q1(mod p) and 2^((q1)/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 q1p. 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