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We know that Goldbach wrote Euler, saying every integer greater than or equal to $6$ is the sum of three prime numbers. Euler responded by saying an equivalent statement is that even integers greater than or equal to $4$ are the sum of two prime numbers.

I need to show why these statements are equivalent. I read some similar posts, but honestly their hints weren't helpful.

Here's what I have so far: Let $p_1,p_2,p_3,p_4,p_5$ be prime numbers. Let $x\geq6\in\mathbb{Z}^+$ and let $y=2d\in\mathbb{Z}$. Now, without losing generality $p_1+p_2+p_3=x$ and $p_4+p_5=y=2d$. If we think of this in terms of divisibility, $(p_1+p_2+p_3)\mid x$ and $(p_4+p_5)\mid2d$, but this seems really trivial. Using congruences, $(p_1+p_2+p_3)\equiv 0$ $\text{mod}$ $x$ and $(p_4+p_5)\equiv 0$ $\text{mod}$ $2$. This seems mildly more helpful, but I'm not sure how to use this to show equivalence between statements.

Running some arbitrary calculations, I notice

$$1+2+3=6, 1+3=4$$ $$2+2+3=7, 3+3=6$$ $$1+2+5=8, 3+5=8$$ $$2+2+5=9, 5+5=10$$ $$5+2+3=10, 5+7=12,$$

and so forth. These representations seem very arbitrary; I don't see how any patters in this can help me. What are some things I should consider or try?

Edit: I forgot to include a few additional thoughts:

For the first statement, if $x$ is even, then the sum must consist of three even primes, or it must consist of two odd primes and one even prime. If $x$ is odd, then it must consist of three odd primes, or it must consist of two even primes and one odd prime.

For the second statement, since $y$ is even, both primes must be even.

I use the convention that even numbers are expressed as $2k$ and odd numbers are expressed as $2k+1$, where $k\in\mathbb{Z}$.

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  • $\begingroup$ $1$ is not a prime number $\endgroup$
    – Peter
    Commented Jun 10, 2021 at 18:53
  • $\begingroup$ I understand that. When Goldbach and Euler had their correspondence, at the time Goldbach considered $1$ as prime number. artofproblemsolving.com/wiki/index.php/Goldbach_Conjecture $\endgroup$
    – user711703
    Commented Jun 10, 2021 at 21:22

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Well, it's clear that Euler's statement implies Goldbach's (since, for an odd number $n$, we can use Euler to represent $n-3=p_1+p_2\implies n=3+p_1+p_2$).

To see that Goldbach implies Euler, take an even integer $n$. By Goldbach we can write $n+2=p_1+p_2+p_3$. By parity, one of the $p_i=2$, say $p_3=2$. But then $n=p_1+p_2$ and we are done.

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  • $\begingroup$ This is so elegant and concise. Thank you so much. Why do you use $n-3$ and $n+2$ to represent odd and even integers? Is this arbitrary? $\endgroup$
    – user711703
    Commented Jun 10, 2021 at 21:24
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    $\begingroup$ The $n-3$ is somewhat arbitrary. For odd $n$ we need to subtract an odd prime to get to an even number. But $n+2$ is necessary, since we are counting on cancelling that $2$. $\endgroup$
    – lulu
    Commented Jun 10, 2021 at 23:29
  • $\begingroup$ That makes sense, I'm grateful for your explanation. Thanks again! $\endgroup$
    – user711703
    Commented Jun 11, 2021 at 20:36
  • $\begingroup$ Additionally, should we also consider a case where $n$ is odd for the Goldbach implies Euler case? $\endgroup$
    – user711703
    Commented Jun 11, 2021 at 20:52
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    $\begingroup$ Here, Euler's claim only applies to even integers. Specifically: Euler's claim is "any even integer $≥4$ can be written as the sum of two primes." (this is what is commonly know as Goldbach's conjecture today) while, again in your question, Goldbach's claim is "any integer $≥6$ can be written as the sum of three primes". $\endgroup$
    – lulu
    Commented Jun 12, 2021 at 8:53

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