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I have an intuition that Average of twin prime pairs is always Abundant number except for 4 and 6. For example:

12 < 1+2+3+4+6=16

18 < 1+2+3+6+9=21


But I can't prove this. Could you give me any good idea?

2010-08-22 I think that Any prime is a factor of average of twin prime pair. Do you agree with me?

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Questions about things like abundant numbers (which fall under the category of "recreational mathematics") are probably best asked at math.stackexchange.com. –  Andy Putman Aug 20 '10 at 2:56
What are "things like abundant numbers"? Specifically, how do you differentiate between important concepts and recreational ones? –  muad Aug 20 '10 at 11:09
I agree this is a genuine mathematical question, but there is a general instinct to close a question like this. I should explain the reason. This question has been around for over 2000 years. The Ancient Greeks probably thought about this question. I personally don't know the answer, but one of the two following possibilities hold: 1) This question can be answered by elementary methods known before Euler's time. In this case, this is not a research level question. –  Alexander Woo Aug 20 '10 at 21:02
2) This question requires algebraic or analytic number theory methods. (For example, a proof might require use of the Riemann zeta function.) In this case, the poser of the question has indicated no familiarity with any of these methods. (Arguably, this still might not make it a research level question, depending on the depth of the methods required.) –  Alexander Woo Aug 20 '10 at 21:07
@a-boy, not much point in tacking a new question on to one that has been closed, as no one can post an answer to a closed question. Then again, there may not be much point in posting it as a new question, either, as an affirmative answer would settle the twin primes conjecture, a negative answer is unlikely, and whether anyone agrees with you or not is not what MO is about. –  Gerry Myerson Aug 22 '10 at 12:43
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closed as off topic by Akhil Mathew, Andy Putman, Qiaochu Yuan, Robin Chapman, S. Carnahan Aug 22 '10 at 7:30

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1 Answer

up vote 9 down vote accepted

Prove the sum is always a multiple of 6, then prove that multiples of 6 are abundant.

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Thank you! I see. –  user8140 Aug 20 '10 at 2:40
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