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Special Primes And Some Of Their Properties
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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.
Title: Special Primes And Some Of Their Properties
Description:
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.
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