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The strong Goldbach conjecture postulates that every even number can be expressed as the sum of two primes. An even number that is just twice a prime is considered a valid exemplar. For low values of n, I cannot find any example of a number that is not the sum of two distinct primes. Among the integers that have been looked at, are there any instances of an integer 2p that can only be expressed as p + p?

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Since 1 isn't counted as a prime, 6 is such an example. –  user29743 Apr 23 '12 at 17:32
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Sorry for being a bit unclear; unusual things happen with low integers, and 4 is formally another example. I was interested in larger integers, say p > 3. 2 and 3 are weird in the context of the set of all primes in that 2 is even and 3 is the only odd prime not expressible as 6k+-1. –  Keith Backman Apr 23 '12 at 18:15
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How far did you look? –  draks ... Apr 23 '12 at 21:57
    
Since I'm not a computer programmer, I'm working by hand and I've looked up to several hundred. But Wikipedia says that Goldbach has been confirmed up to 10exp18, so somebody has looked by computer methods, and I wondered what they found. –  Keith Backman Apr 23 '12 at 22:09
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2 Answers

Sorry if this is not viewed as an answer but since there is clearly never going to be a definitive answer it's possibly the nearest thing anyone is likely to come up with.

I wrote a program that tested the number of Goldbach Pairs for the primes between 131 and 48623 (the 5000th prime) by which I mean for the Even numbers that are twice each of these primes and here are my results:

  • The minimum number of Goldbach Pairs in addition to [Pn,Pn] (i.e. distinct pairs) over this interval is 6
  • The maximum prime at which this minimum occurs is 199 (2P=398)
  • The distinct pairs corresponding to this are [241,157][271,127][331,67][337,61][367,31] and [379,19]
  • The average number of distinct GB Pairs over this range is 312
  • The average number of distinct GB Pairs increases steadily as the number of primes increases and up to the 2000th prime it is 139

I hope this gives you an idea of just how unlikely it is that the Slightly Stronger Goldbach Conjecture will ever be disproved.

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I saw a writeup years ago that applies here as well. The original Goldbach conjecture has been verified by computer up to some high limit. Wikipedia claims $4 \cdot 10^{18}$ but the exact value is not so important. In the heuristic that the primes are randomly distributed, you can check the chance that it will fail at $N$ by making all decompositions into $N=p+q$ with $p \ge q$ and see if they fail. Your strengthening is to make the inequality strict. Since the chance that a "random number" $p$ is prime is $\frac 1{\ln p}$, the chance that $N$ can be expressed this way as a sum of primes is $\frac 1{\ln p \ln q}$. The chance that $N$ witnesses the failure of the Goldbach conjecture is $\int_2^{\frac N2} \frac {dn}{\ln n \ln (N-n)}$ and the effect of the strengthening is to reduce the upper limit to $\frac N2-1$. The chance that the Goldbach conjecture fails is then $\int_N^\infty \int_2^{\frac N2} \frac {dn}{\ln n \ln (N-n)}$ where $N$ is the lowest unchecked number. This yields a very small probability. The probability for the strengthened version will be higher because $N$ is much lower. The impact of the strengthening will be very small. This all works because even numbers have many ways of being expressed this way.

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