I want to see a simple proof of a theorem that is weaker than chen's theorem.
Thus let $m,n$ be positive integers. An m-almost prime is a squarefree integer that is the product of at most $m$ primes. ( I added the squarefree condition to the " definition " )
Prove that there are an infinite amount of numbers $n,n+2$ such that both $n$ and $n+2$ are m-almost primes.
The case $m=2$ is known as Chen's theorem , or almost since I added the squarefree condition.
the case $m=1$ is the unsolved prime twins conjecture.
Since the proof of Chen's theorem is too complicated for a beginner like me , It seems larger $m$ should be simpler.
So I want to see a simple proof for a large $m$ so that I can get a first step into " this kind of number theory ".