# Could Someone Please Verify My Proof? (fun with two primes)

Am I wrong? Is my proof clear? Is this already a known result? If so, are there any simple implications or corollaries of this proof? (if that makes sense) I know some about groups, rings and fields, but not enough to argue with them.

Suppose $$n,k\in\mathbb{N}$$ such that $$n=2k$$, and $$k>2$$.
$$\exists x,y\in\mathbb{N}$$ such that $$x+y=k-1$$.
I claim that if $$2x+1$$ and $$2y+1$$ are prime then $$n$$ is the sum of two primes.

\begin{align*} n&=2k,\\ &=2(x+y+1),\\ &=4xy+2(x+y+1)-4xy,\\ &=4xy+2x+2y+2-4xy,\\ &=(4xy+2x+2y+1)+1-4xy,\\ &=(4xy+2x+2y+1)+1-((2x+1)-1)((2y+1)-1),\\ &=(2x+1)(2y+1)+1-((2x+1)-1)((2y+1)-1). \end{align*}

Let $$m,l\in\mathbb{N}$$ such that $$m=2x+1$$ and $$l=2y+1$$.

Then $$n=ml+1-(m-1)(l-1)$$ and

$$\forall p_{1},p_{2}\in\mathbb{N}$$ such that $$p_{1}$$ and $$p_{2}$$ are prime, if $$m=p_{1}$$ and $$l=p_{2}$$
\begin{align*} n&=2k,\\ &=ml+1-(m-1)(l-1),\\ &=p_{1}p_{2}+1-(p_{1}-1)(p_{2}-1),\\ &\therefore n=p_{1}+p_{2} \end{align*}

• Welcome to MSE. Proof of what? Mar 7, 2021 at 7:12
• The question shows some effort even if the result turned out to be trivial so I don't think downvotes are warranted. Mar 7, 2021 at 7:24
• lol and thank you. I admit I don't really know what I'm talking about and that I am basically repeating myself. It seems like the way I arrived at the result is worth looking at though. I just wondered if anyone knew of any methods of proof that somehow utilized this, or about any results that seemed apparent from looking at the sum of two primes this way
– matt
Mar 7, 2021 at 7:39
• "I just wondered if anyone knew of any methods of proof that somehow utilized this" You mean algebraic manipulation? If so then it has been utilized extensively. Mar 7, 2021 at 7:55
• That's a roundabout derivation of $\ 2(x+y+1) = 2x\!+\!1\,+\, 2y\!+\!1\ \$ Mar 7, 2021 at 8:26

I'm afraid the result is quite trivial: $$x+y=k−1$$ $$2x+2y=2k-2=n-2$$ $$(2x+1)+(2y+1)=n$$ So if $$2x+1$$ and $$2y+1$$ are prime then $$n$$ is the sum of two primes.