On arithmetic terms expressing the prime-counting function and the n-th prime

TL;DR

This paper introduces fixed-length elementary closed-form arithmetic expressions for the prime-counting function and the n-th prime, using only basic arithmetic operations, providing a constructive approach to understanding prime order.

Contribution

It develops the first fixed-length elementary arithmetic terms for both $\pi(n)$ and $ ext{ extit{p}}(n)$, advancing the explicit representation of prime-related functions.

Findings

Fixed-length elementary expressions for $\pi(n)$ and $ ext{ extit{p}}(n)$ are constructed.

Arithmetic term for the prime omega function $\omega(n)$ is developed.

New prime-related exponential Diophantine equations are introduced.

Abstract

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $\omega(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $\pi(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes?