Properties and applications of the prime detecting function: infinitude of twin primes, asymptotic law of distribution of prime pairs differing by an even number

TL;DR

The paper introduces the prime detecting function (PDF) approach, demonstrating its effectiveness in proving the infinitude of twin primes and the Hardy-Littlewood conjecture for prime pairs, and exploring asymptotic prime pair distribution.

Contribution

It presents a novel PDF method that simplifies prime number investigations, proving twin primes are infinite and addressing the distribution of prime pairs.

Findings

Proof of the infinitude of twin primes

Validation of the Hardy-Littlewood conjecture for prime pairs

Introduction of a new prime detecting function approach

Abstract

The prime detecting function (PDF) approach can be effective instrument in the investigation of numbers. The PDF is constructed by recurrence sequence - each successive prime adds a sieving factor in the form of PDF. With built-in prime sieving features and properties such as simplicity, integro-differentiability and configurable capability for a wide variety of problems, the application of PDF leads to new interesting results. As an example, in this exposition we present proofs of the infinitude of twin primes and the first Hardy-Littlewood conjecture for prime pairs (the twin prime number theorem). On this example one can see that application of PDF is especially effective in investigation of asymptotic problems in combination with the proposed method of test and probe functions.