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Prove that for all odd prime numbers $p_1$ and $p_2$, there exist prime numbers(exclude 2) $p_3$ and $p_4$ such that $$p_3 + p_4 = p_1 + p_2 + 2.$$

Hints would be appreciated.

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    $\begingroup$ What is the source of this problem? $\endgroup$ – user236182 Nov 8 '15 at 0:49
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If this is true then the Goldbach conjecture is true. It is famous, and unproved.

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    $\begingroup$ Start with 2+2=4. Then, by induction, every even number greater than two is the sum of two prime numbers. $\endgroup$ – Empy2 Nov 7 '15 at 16:51
  • $\begingroup$ Well prove it then. $\endgroup$ – user85798 Nov 7 '15 at 17:00
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For $p_1=2,p_2=7$, we have $p_1+p_2+2=11$, but there are no pairs of primes $(p_3,p_4)$ such that $$11=p_3+p_4.$$

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    $\begingroup$ Ah damn I forgot to exclude 2. Respect. $\endgroup$ – user85798 Nov 7 '15 at 17:14
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    $\begingroup$ It's better to light a candle than to curse the darkness, and it's better to edit your question than to lament on what you should have said in the first place. $\endgroup$ – Scott Nov 7 '15 at 20:11

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