I'm reading a paper regrading the basis orders. In that paper, I met with the following statement:

$$3(\mathbb{P}\cup\{0 \})=\mathbb{Z}_{\geq 2}$$,

Which, by definition, states that primes form basis of order $3$(Must admit that I still don't really know what it really means).

My question is, why the statement "every integer greater than $2$ can be represented as sum of three prime" equivalent to Goldbach's conjecture.

  • 1
    $\begingroup$ Because take an even number $n$, if $n+2$ is the sum of three primes, then one of those primes must be even, and hence is $2$. It follows that $n$ is itself the sum of $2$ primes. $\endgroup$ – Joel Moreira Aug 23 '15 at 17:55
  • $\begingroup$ Hello. That's case is obvious, but what happens with the case in which $n$ is odd integer? $\endgroup$ – russiankingdom Aug 23 '15 at 18:23
  • $\begingroup$ I would interpret it this way : Every natural number $n>1$ can be represented by the sum of three numbers, being either a prime or $0$. If exactly three numbers are required (not at most three numbers), then $0$ must be added becuase $5$ would otherwise not have a representation. $\endgroup$ – Peter Aug 23 '15 at 18:23
  • $\begingroup$ Yes. But why it equivalent to Goldbach's conjecture? $\endgroup$ – russiankingdom Aug 23 '15 at 18:24
  • $\begingroup$ @russiankingdom Then use Golbach to write $n-3$ as sum of two primes. $\endgroup$ – Joel Moreira Aug 23 '15 at 18:26

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