On Primes of the Form 2x-q (where q is a prime less than or equal to x) and the Product of the Distinct Prime Divisors of an Integer (Revised): A Function Approach to Proving the Goldbach Conjecture by Mathematical Induction

This paper is really an attempt to solve the age-old problem of the Goldbach Conjecture, by restating it in terms of primes of the form 2x-q (where q is a prime less than or equal to x). Restating the problem merely requires us to ask the question: Does a prime of form 2x-q lie in the interval [x, 2x]? We begin by introducing the product, m, of numbers of the form 2x-q. Using the geometric series, an upper bound is estimated for the function m. Next, we prove a theorem that states every even number, 2x, that violates Goldbach’s Conjecture must satisfy an inequality involving a simple multiplicative function defined as the product, ρ(m), of the distinct prime divisors of m. A proof of the Goldbach Conjecture is then evident by contradiction as a corollary to the proof of the inequality.

本文旨在求解哥德巴赫猜想（Goldbach Conjecture）这一古老难题，通过将其重构为形如2x−q的素数的表述形式（其中q为不大于x的素数）。该问题的重构仅需提出如下设问：形如2x−q的素数是否落在区间[x, 2x]内？
本文首先引入形如2x−q的整数的乘积m，并借助几何级数对函数m的上界进行估计。随后，本文证明了一条定理：所有不满足哥德巴赫猜想的偶数2x，必然满足一个涉及某类简单乘性函数的不等式，该函数被定义为m的不同素因子的乘积ρ(m)。最后，通过反证法，以该不等式的证明为推论即可直接推导出哥德巴赫猜想的证明。