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.
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.