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It is well known fact that it is very hard to prove Goldbach's strong conjecture but perhaps some weaker variations can be proved(or disproved) ,so my question is: Is it true that every even number greater than 10 can be represented as the sum of an odd prime number and an odd semiprime?

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@yunone,Thanks... –  pedja Sep 13 '11 at 8:58
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Have you seen Chen's theorem (en.wikipedia.org/wiki/Chen%27s_theorem) ? I don't know if that can be extended to just the prime + semiprime case, though –  yatima2975 Sep 13 '11 at 9:01
    
@yatima2975: That's probably as good an answer as this question will get here. (I was going to post it myself before I rechecked the comments.) –  anon Sep 13 '11 at 9:05
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I wrote a small c++ program that verified the conjecture for all even numbers less than 600k and larger than 10. So perhaps it's true. –  JSchlather Sep 13 '11 at 9:12
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math.utoledo.edu/~jevard/Page015.htm Has some references regarding improvements on Chen's theorem. –  JSchlather Sep 13 '11 at 11:57

1 Answer 1

Some counterexamples: 12, 14, 16, 30. My perl program can't find any more smaller than 100000.

EDIT: I didn't know that semiprimes are defined to include squares. When I comment out the line that filters them, nothing is output up to 100000. I'll leave this answer here as an example of wrongness.

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9+3=12, 9+5=14, 9+7=16, 25+5=30. Are you forgetting that $p^2$ is semiprime in your code? –  JSchlather Sep 13 '11 at 9:16
    
@Dan,12=3+9,9 is odd semiprime ;14=5+9 ;16=7+9; 30=5+25 ,25 is odd semiprime...so you are not right this time –  pedja Sep 13 '11 at 9:18

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