A Necessary Condition for Determining the Validity of Twin Prime Pairs
Dasheng Liu *
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai, China.
*Author to whom correspondence should be addressed.
Abstract
The distribution of prime numbers in natural number sequences is extremely irregular, appearing densely at times and far apart at times. Two prime numbers with a spacing of 2 are called twin prime numbers, and where such prime pairs appear in natural number sequences is one of the key issues that urgently needs to be addressed regarding the distribution of prime numbers. Although some progress has been made in finding twin prime numbers with the improvement of the Eratosthenian sieve method and other computing algorithms, there is still a lack of theoretical and efficient methods for determining twin prime pairs. The main aim of this paper is to explore the establishment of a necessary condition for determining whether any set of adjacent odd numbers is a twin prime pair. Firstly, based on Wilson’s theorem, we derive a congruence equation for the n−th power of 2 over a given modulus of (2n+1). Then, a novel necessary condition is obtained for judging the validity of twin prime pairs by using the Chinese remainder theorem. Finally, some computational examples are provided to demonstrate the effectiveness of the proposed method.
Keywords: Twin prime pairs, necessary condition, congruence relationship, n-th power of 2