Updated: Definitive Proof of the Twin Prime Conjecture скачать в хорошем качестве

Updated: Definitive Proof of the Twin Prime Conjecture

A twin prime is defined as a pair of prime numbers $(p_x,p_y)$ such that $p_x + 2 = p_y$. The Twin Prime Conjecture states that there are an infinite number of twin primes. A more general conjecture by de Polignac states that for every natural number $k$, there are infinitely many primes $p$ such that $p + 2k$ is also prime. The case where $k = 1$ is the Twin Prime Conjecture. In this document, the function $\Pi_2^*(n)$ is derived that closely approximates $\Pi_2(n)$, the actual number of twin primes less than $n$, for large values of $n$. Then by proof by induction on $\Pi_2^*(n)$ , it is shown that for any prime number $p_i$, there is at least one twin prime ($p_x,p_y$) such that $p_i^2$ is less than $p_y$ is less than $ (p_{i+1})^2$. Since there are an infinite number of prime numbers $p_i$, this proves that there are an infinite number of twin primes, thus proving the Twin Prime Conjecture. Error analysis shows that the maximum error between $\Pi_2^*(n)$ and $\Pi_2(n)$ increases at a slower rate than $\Pi_2^*(n)$. Using this same methodology, the de Polignac Conjecture is also shown to be true.