# Equivalence between Goldbach and Euler's statements

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 $$k\in\mathbb{Z}$$.

• $1$ is not a prime number Jun 10, 2021 at 18:53
• 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
– user711703
Jun 10, 2021 at 21:22

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.

• 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?
– user711703
Jun 10, 2021 at 21:24
• 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 $2$.
– lulu
Jun 10, 2021 at 23:29
• That makes sense, I'm grateful for your explanation. Thanks again!
– user711703
Jun 11, 2021 at 20:36
• Additionally, should we also consider a case where $n$ is odd for the Goldbach implies Euler case?
– user711703
Jun 11, 2021 at 20:52
• 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".
– lulu
Jun 12, 2021 at 8:53