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?
Even numbers greater than 10 as sum of two specific odd numbers
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number-theory
prime-numbers
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4Have you seen Chen's theorem (http://en.wikipedia.org/wiki/Chen%27s_theorem) ? I don't know if that can be extended to just the prime + semiprime case, though – 2011-09-13
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0@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.) – 2011-09-13
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0@yatima2975,Chen's theorem states for every sufficiently large even number while my question is more specific since it states for each even number greater than 10..... – 2011-09-13
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0pedja: It's apparent in yatima's comment that he knows that. – 2011-09-13
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3I 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. – 2011-09-13
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2http://www.math.utoledo.edu/~jevard/Page015.htm Has some references regarding improvements on Chen's theorem. – 2011-09-13
1 Answers
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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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19+3=12, 9+5=14, 9+7=16, 25+5=30. Are you forgetting that $p^2$ is semiprime in your code? – 2011-09-13
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0@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 – 2011-09-13