Let's take even number $100$ as an example (an example in the paper):
Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, n<2,n \neq 100 \mod 2\}=\{1\}\\ &B=\{n: n \in \mathbb{Z^+}, n<3,n \neq 100 \mod 3\}=\{2\}\\ &C=\{n: n \in \mathbb{Z^+}, n<5,n \neq 100 \mod 5\}=\{1,2,3,4\}\\ &D=\{n: n \in \mathbb{Z^+}, n<7,n \neq 100 \mod 7\}=\{1,3,4,5,6\}\\ &E=\{n: n \in \mathbb{Z^+}, n<100,n \mod 2 \in A,n \mod 3 \in B,n \mod 5 \in C,n \mod 7 \in D\}=\{11, 17, 29, 41, 47, 53, 59, 71, 83, 89\} \end{align*} $$ We see that elements of the set $E$: $\ 11+89=100,\ 17+83=100, \ 29+71=100,\ 41+59=100, \,47+53=100$
Why elements of the set $E$ can be Goldbach pairs for the given even number 100?
Maybe there's an answer in that paper, but it may take pages and in bad math, hope there's an answer of coolness.Any revise,modify,suggestion,edit are welcomed.
Fix some large even $n$. The paper constructs the set $$A_p = \{0 < m \leq n\,|\, m\not\equiv 0, n\!\pmod{p}\}$$ for each prime $p\leq \sqrt{n}$. Now, if $1 < m < n$, then $m$ is certainly prime if $m\in A_p$ for all $p\leq \sqrt{n}$. We want to find such $m, m'$ with $m + m' = n$. Long story short, the paper then uses the Chinese remainder theorem to show that $$A = \bigcap_{p\leq \sqrt{n}} A_p$$ is nonempty. (Note that $n$ is even, so $A_2$ is nonempty.) That doesn't follow. There's certainly an element $d$ that satisfies the given system of congruences, but there's no reason why it should be on the interval $[0, n]$. (It's only determined mod $N = \prod_{p\leq \sqrt{n}} p$, which is generally much larger. Much larger, in fact; for $n = 10^4$, the product is more than $10^{36}$.) Without that criterion, it doesn't follow that $d$ is prime.
I think that's what the paper is claiming, anyway. It's very badly written, and it has all the classic hallmarks of bad amateur math: Word instead of LaTeX, being distributed outside of a journal or something like the arxiv, poor writing quality, obsessive detail over trivial examples and completely elementary results (we know about the Chinese Remainder theorem; you don't have to spend two pages discussing it), exactly one reference, and no references to papers or anything beyond introductory textbooks. I really don't want to be that guy (and I hope someone with more patience than me will dissect the paper in more detail), but this is not good math.