Generation and Distribution of Prime Numbers Using a Modified Lagrange Polynomial

A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating algorithm based on a 1-degree polynomial that can generate prime numbers from consecutive primes. The algorithm is based on the condition that infinitely many prime numbers exist that satisfy the equation $\Pi_{n} =2\Pi_{n-1} - \Pi_{n-2} \pm 2 \ \ \forall \ \Pi_{n} >7$. where $\Pi_{n-1}$ and $\Pi_{n-2}$ are the consecutive primes.