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My number theory textbook has the following (paraphrased) exercise:

Goldbach wrote a letter to Euler with the following conjecture: Every integer greater than five can be written as the sum of three primes.

Euler replied this statement was equivalent to: Every even integer greater than or equal to 4 can be written as the sum of two primes.

The exercise is to show these two statements are equivalent, and I'm stuck on this. I don't want a solution but nothing is coming to me, so I was just hoping for a hint.

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2 Answers 2

up vote 8 down vote accepted

(Goldbach $\implies$ Euler)

Suppose that $n\geq 6$ (i.e. $n>5$) is even. Express $n$ as a sum of three primes $$n=p_1+p_2+p_3.$$ What do you think one of the primes must be, if we're going to conclude that any even integer $\underline{\,\,\,\mathbf{\geq 4}\,\,\,\,\,}$ is a sum of two primes? Modular arithmetic will help you prove that this is one of the three primes.


(Euler $\implies$ Goldbach)

Suppose that $n\geq 6$. Consider $n-2$ or $n-3$ depending on whether $n$ is even or odd.

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For the second case, you only need to consider $n$ odd and $n-1$ in this case. –  lhf Sep 20 '11 at 2:18
    
@lhf: I'm afraid I don't understand your comment; $0$ and $1$ are not primes. –  Zev Chonoles Sep 20 '11 at 2:19
    
@lhf, why? you do need an extra prime to show up there. –  Soarer Sep 20 '11 at 2:19
    
I like both of your answers, but you answered first, so I'll accept this one. Thanks a lot. It was a lot easier than I expected. –  mathmath8128 Sep 20 '11 at 2:25
1  
Ah, my reduction to $n$ odd was due to a misreading of Goldbach's formulation as at most three primes. –  lhf Sep 20 '11 at 2:28

HINT:

Let us consider the first few numbers considered in both the weak and strong form. For n = 6, they both say that it can be hit. For n = 7, the weak form says it can be hit, and it's 3 more than 4. For 8, both. For 9, the weak says it can be hit, and it's 3 more than 6 (an even number).

That's funny. 3's a prime, isn't it?

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Okay, I like this. :-) –  mathmath8128 Sep 20 '11 at 2:19

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