This is one of my favorite youtube channels. They make videos on various math topics that are understandable by laypeople.
One of the underlying challenges for number theory is closing the gap between the great many statements that we can easily predict on the basis of primes acting a lot like random numbers, and the statements that we know how to prove. Both the Riemann hypothesis and the twin prime conjecture are good examples.
These constellations provide more examples of the same. We can in a straightforward way rule out constellations that can only happen a finite number of times. And for those which can happen an infinite number of times, we can predict the frequency with which they will happen.
Should any such constellation happen a statistically unlikely amount given that prediction, this would be of great interest for number theorists. Unfortunately to date they have stubbornly behaved as predicted, but that doesn't mean that the effort spent searching was wasted.
A lot of cryptography used to be just number theory until computers came along and were powerful enough to make use of it. How to tell if something if someting is divisible by 3. Checksums as used on credit cards. Euler's algorighm.
No, sexy primes don't really have a point other than they are identified and there is probably some unused conjecture that they are infinite in number.
The analogy I'd give is from physics. Understanding the prime number structure is like understand how an arbitrarily complex 3-dimensional shape will interact with another equally arbitrarily 3-dimensional complex shapes (let's just assume rigid-body interaction here).
But it should be intuitively obvious that starting with the question you want to answer "how do arbitrarily complex shapes interact" (the analog, in our example, to "how are arbitrary primes structured") is too big an undefined question to answer directly. Maybe somebody will be able to do it, but most likely it will be solved by breaking it into smaller, incomplete, but accurate models that though comparing and contrasting (e.g. why do circles interact differently than squares) and combination (e.g. I know circles interact, I know how squares interact, I can now define a grand circle/square unification theory that describes how circles and squares interact) .
So, you break the problem down into questions like "how do circles interact?", "how do squares interact?", "how do one-dimensional shapes interact?", "2D?". By identifying subclasses of the overall uber problem it's possible to solve a hard larger problem.
Back to the primes example, each different metric for defining a relationship between primes effectively defines a new class of primes that can be probed to figure out why they act in the way they do and how they are distributed. Each class of prime is a (probably, but not necessarily) incomplete yet accurate model for how all primes operate overall.
Probably just a sense of humor
You can also compare their density with normal primes on this page - https://prime-numbers.info/special/visual-type-comparison#se...
Best thing is that you can enjoy those videos for 11 hours. :)
Lucky Prime etc...
More generally, see the article on prime gaps: https://en.m.wikipedia.org/wiki/Prime_gap
That doesn't match what the article says: "prime numbers that differ from each other by six". So they're separated by five other numbers (which are not necessarily all non-primes).
Great to see the HN tradition of downvoting for an off by one error continues strong as well...