Goldbach's Conjecture Revisited

I touched on Goldbach's Conjecture in a post from November of 2015 but I was reminded of it once again when I celebrated my 25156th day on Earth. Here is my tweet for the day:

For any given number, WolframAlpha will return the prime decomposition with the lowest prime. It doesn't say so explicitly but a few test numbers reveal that this is what's happening e.g. 100 returns 3 and 97. Here is a screenshot of what happens to 50312 which is 25156 x 2:

Let's remember that the Goldbach conjecture states that every even number can be expressed as the sum of two primes. OEIS sequence A001031 shows how many compositions are possible for numbers up to 10,000. For 10,000 there are 231 possible compositions and WolframAlpha as said will quickly return the one involving the smallest primes, namely 59 and 9941. 

A plot of the number of compositions against the even integers themselves, sometimes called Goldbach's comet, is shown below for numbers up to 2000:

While there is clearly a general trend toward larger numbers of compositions as the numbers increase in size, there is little evidence of this at the local level. For example, here is a sample of the number of compositions for odd and even integers between 9975 and 10,000 (even integers have their number of compositions marked in bold type):

9975 559, 9976 165, 9977 183, 9978 338, 9979 175, 9980 217, 9981 330, 9982 215, 9983 174, 9984 357, 9985 225, 9986 165, 9987 331, 9988 187, 9989 213, 9990 454, 9991 164, 9992 163, 9993 335, 9994 180, 9995 219, 9996 435, 9997 192, 9998 167, 9999 366, 10000 231

Of course, what OEIS A208662 is saying is that 25156 is the smallest number in which it's double (50312) has the prime number 181 appearing for the first time as one of the two primes whose sum is equal to 50312. This is not so easy to show unless all the compositions less than 50312 are tested. OEIS A002373 provides a list of the smallest prime number in the composition of all integers up to 20,000 but that falls a little short of 25156. Scanning through the list, it's clear that most of the primes are quite small and the bigger primes make only relatively rare appearances. For example, 89 doesn't make an appearance until 19828. I'll leave off here as I've already spent a lot of time dithering around with this.

ADDENDUM: added on November 24th 2019

It's embarrassingly easy to find the minimal Goldbach composition for a number using only a table of primes. The smaller of the two primes is generally very small and so only needs to be subtracted from the number. If the result of the subtraction is a prime number, then you have your minimal Goldbach decomposition. In the case of 50312, you only need to test up to 181 before you succeed. Here is some SageMath code that does this:

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