Distribution of Prime Numbers and Fibonacci Polynomials

Squares of odd index Fibonacci polynomials are used to define a new function \(\Phi\left(10^{n}\right)\) to approximate the number \(\pi\left(10^{n}\right)\) of primes less than \(10^{n}\). Multiple of 4 index Fibonacci polynomials are further used to define another new function \(\Psi\left(10^{n}\right)\) to approximate the number \(\Delta\left(\pi\left(10^{n}\right)\right)\) of primes having \(n\) digits and compared to a third function \(\Psi’\left(10^{n}\right)\) defined as the difference of the first function \(\Phi\left(10^{n}\right)\) based on odd index Fibonacci polynomials. These three functions provide better approximations of \(\pi\left(10^{n}\right)\) than those based on the classical \(\left(\frac{x}{log\left(x\right)}\right)\), Gauss’ approximation \(Li\left(x\right)\), and the Riemann \(R\left(x\right)\) functions.