Prime Number Sums and Counts: A Unified Approach via Inclusion-Exclusion

This repository contains two versions of the same paper: - **Portuguese (original):** "Soma e Contagem de Números Primos: Uma Abordagem Unificada via Inclusão-Exclusão" - **English (translation):** "Prime Number Sums and Counts: A Unified Approach via Inclusion-Exclusion" The paper develops a discrete algebraic framework for the study of prime numbers, starting from the sum of natural numbers. From the identity \(SPr(x) = 1 + x^2 - SC_I(x)\), we derive closed formulas for the exact sum and the count of primes in the interval \([1, 2x]\), using the inclusion-exclusion principle over multiples of odd primes. Main results include: - A closed formula for the prime sum \(SPr(x)\); - A closed formula for the prime count \(QPr(x) = π(2x-1)\); - A primality criterion \(Δ(x) = SC_I(x+1) - SC_I(x)\) that requires no divisions; - An indicator function \(f(x)\) and a generating function \(G(x)\) that returns the prime itself or zero; - Connections to Goldbach's Conjecture, twin primes, and gaps between consecutive primes; - Algorithmic recipes for computational implementation of sum and count. All proofs are elementary, finite, and avoid logarithms, trigonometric functions, or asymptotic approximations. The construction is entirely discrete and verifiable for any finite interval.