Goldbach’s conjecture states that every even number greater than 2 is the sum of two prime numbers. For example 4=2+2, 10=3+7, and 194=13+181.

The thing is, as 2n becomes large, it has not just one but many pairs of primes that can be added to produce it. Even the 10=3+7 example above also has 10=5+5.

A scatter plot can show how the number of prime pairs consistently grows, but also the large and curiously patterned variation in this growth. This is known as Goldbach’s comet.

In the render above, I used color to indicate density — from multiple points landing on the same pixel. This occurs because there are 10,000,000 numbers on the x axis but the image is only 2,560 pixels wide. The color variation is automatically scaled so that each part of the spectrum from red (only one point landed on the pixel) to purple (very many points landed on the pixel) is used equally.

I was inspired to do this because when looking for renders of Goldbach’s comet none offered a way to see this density, and so much less detail is seen, such as this one. I used a web app to create it, source code here. More information on Golbach’s comet on Wikipedia.