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 ${\mathrm{\Pi }}_n=2{\mathrm{\Pi }}_{n-1}-{\mathrm{\Pi }}_{n-2}\pm 2\mathrm{\forall }{\mathrm{\Pi }}_n>7$. where ${\mathrm{\Pi }}_{n-1}$ and ${\mathrm{\Pi }}_{n-2}$ are the consecutive primes.