# Even numbers greater than 6 as sum of two specific primes

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
@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
pedja: Because Goldbach immediately follows from it, but not vice versa. @Gerry: Good point. –  anon Sep 13 '11 at 7:25

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
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
Dan, sorry for the editing inconvenience, race conditions I suppose... –  Ｊ. Ｍ. Sep 13 '11 at 8:51
J. M., I submitted it and it is under review. oeis.org/A195324 –  Dan Brumleve Sep 15 '11 at 6:13