My wife and kids were hiking up in New Hampshire for the weekend and I just let a simple Goldbach Comet program run in Mathematica while they were gone. Here’s a version of that program. The graph shows the number of ways to write the even integer 2N (up to 100,000) as a sum of two primes. I forgot to label the x-axis correctly, which is why the last label is 50,000 rather than 100,000.

One thing I thought would be fun to do was to look at the numbers that can be written in many different ways as the sum of two primes – so the very top of this graph. I got an bit of a surprise:

So, what this picture is showing is the number with the highest number of partitions. So 330,330 sets the record with 6,181 partitions into two primes. this record isn’t broken until 351,120 which has 6,363 different partitions.

The surprise is that there are so many numbers that are multiples of 1,001. What is it about these numbers that leads to so many different ways to write them as a sum of two primes?

A look at the factorization of the last 5 numbers suggests that these numbers have really simple factorizations (to help read this chart, the first number is factored as follows:

It wouldn’t surprise me at all if there’s a relatively easy explanation for what’s going on here – but I don’t see it! Why would numbers that factor nicely have lots and lots of ways to be written as sums of two primes?

The next numberphile video on 210 kind of explains how this nature of prime factorization helps to see all primes between 105 and 209 are one summand q of p+q and 210-q = p also is prime.

That all primes bigger than n/2 are one summand does not happen with any number bigger than 210, but even if this does not apply to all primes within [n/2, n] where n>210 and n has such a factorization, I guess more of these primes will be part of a solution.

I understand the concentration on all the primes in [n/2,n] and not in [1,n/2] is because primes become more sparse and so the upper half primes are better candidates for the possible summand pairs than are the lower half primes. So concentrating on that interval makes sense.

## Comments

The next numberphile video on 210 kind of explains how this nature of prime factorization helps to see all primes between 105 and 209 are one summand q of p+q and 210-q = p also is prime.

That all primes bigger than n/2 are one summand does not happen with any number bigger than 210, but even if this does not apply to all primes within [n/2, n] where n>210 and n has such a factorization, I guess more of these primes will be part of a solution.

I understand the concentration on all the primes in [n/2,n] and not in [1,n/2] is because primes become more sparse and so the upper half primes are better candidates for the possible summand pairs than are the lower half primes. So concentrating on that interval makes sense.