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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 ,so my question is: Is it true that every even number greater than 6 can be represented as the sum of an odd prime number and an safe prime?

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That's actually a stronger variation. –  anon Sep 13 '11 at 7:21
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@anon, yes, but I suppose that means that, if it's false, it would be easier to prove it's false. pedja, care to remind us what a "safe prime" is? –  Gerry Myerson Sep 13 '11 at 7:23
    
@anon,Why do you think so? –  pedja Sep 13 '11 at 7:24
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pedja: Because Goldbach immediately follows from it, but not vice versa. @Gerry: Good point. –  anon Sep 13 '11 at 7:25
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up vote 6 down vote accepted

No. For example, 32 is not the sum of an odd prime and a safe prime. This is because the only safe primes smaller than 32 are 5, 7, 11, and 23, and we have:

$\begin{align*} 32&= 5 + 27\\ 32&= 7 + 25\\ 32&= 11 + 21\\ 32&= 23 + 9\end{align*}$

I whipped up an inefficient perl program to calculate counterexamples, which include: 32, 56, 92, 98, 122, 128, 140, 152, 176, 194, 212, 224, 242, 254, 260, 272, 296, 302, 308, 326, 332, 368, 392, 398, 410, 422, 434, 452, 458, 476, 488, 500, 512, 518, 524, 536, 542, 560, 572, 596, 602, 632, 644, 656, 662, 674, 686, 692, 704, 710, 728, 752, 770, 782, 788, 800...

Based on the heuristic justification of Goldbach's conjecture and the assumption that the primality of odd $k$ and the primality of $\dfrac{k-1}{2}$ are independent, I would conjecture that there are only a finite number of such counterexamples. The expected number of solutions should be about $\dfrac{n}{2\log(n)^3}$. The same argument applies even if both primes are safe, giving about $\dfrac{n}{2\log(n)^4}$ solutions. But the experimental evidence is not very convincing, so I wonder if there is some flaw in this argument? Also, I'm suspicious of the fact that the program outputs $32$, $128$, $512$, and eventually $2048$ as well. Is there some reason that if the sum of two odd primes is an odd power of two, neither of them can be safe?

EDIT: André Nicolas has shown in his answer to my follow-up question that there are an infinite number of exceptions to this claim.

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,20=13+7, where 7 is safe prime –  pedja Sep 13 '11 at 7:34
    
Oops! I'll look for another one... Found it! pedja, I'm sorry for being wrong (again). –  Dan Brumleve Sep 13 '11 at 7:34
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The sequence $32,56,92\dots$ isn't in the OEIS. Maybe you or somebody else might want to submit it... :) –  J. M. Sep 13 '11 at 8:33
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Dan, sorry for the editing inconvenience, race conditions I suppose... –  J. M. Sep 13 '11 at 8:51
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J. M., I submitted it and it is under review. oeis.org/A195324 –  Dan Brumleve Sep 15 '11 at 6:13
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