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Given a Diophantine equation $p(x_1,x_2,...,x_n)$,

Can I find a reduction from $\text{dioph}(\mathbb{N}) \leq \text{dioph}(\mathbb{N}_e)$?

$\mathbb{N}_e$ is the set of even numbers.

So I have to find a manipulation of $p$ such that I only have even numbers as solution.

How can I do this? Unfortunately I have no Idea.

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So I have to find a manipulation of $p$ such that I only have even numbers as solution.

Not so. You have to find a manipulation of $p$ such that if you can find a solution / all the solutions of the manipulated $p'$ in even numbers, then you can find a solution / all the solutions of $p$. That's a very different problem.

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How about $p(x_1/2,x_2/2,...,x_n/2)$? For example, if we had $x_1^2+x_2^2=5$, we will have $\dfrac{x_1^2}4+\dfrac{x_2^2}4=5$.

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