Today I continued to play with the problem from my post Puzzling Primes. Among other things, I installed the computer algebra system SageMath, which I can use to test quickly whether numbers are prime. With that tool, I might be able to solve the problem by brute force. I can’t imagine that this would be chosen as a problem of the week if there were not a better way, though.
So far I have been able to prove (a) that a superprime may contain at most two occurrences of the digits 1 or 7, (b) that all the other digits must be 3 or 9, except the first digit, which may also be 2 or 5, and (c) that if the first digit is 2 or 5, there will be no occurrences of 1 or 7.
All this means that, if there were some factor limiting the number of trailing 3s and 9s a prime could have, I would at least be able to prove a limit on the length of superprimes, and therefore that there must be a largest one. So far I’ve had no luck with that avenue of inquiry, though.
I will share the actual proofs of these claims in the future, but I’m still feeling unwell today. In the meantime, here is a chart of the ways that potential superprimes can be constructed. Red represents potential starting digits while blue represents digits that may be added.
Wow. I’m pretty intrigued by all this. I’ve never found all these facts.
The primes you’re looking at are often called right-truncatable. I haven’t thought much about them, but I’ve looked at left-truncatable primes, of which there are many more (4260; the largest has 24 digits). With the machinery you’re developing, finding the right-truncatable primes kinda sorta by hand looks more possible than I imagined.
A question: if you do hunt them down systematically, how will you know you are done and can stop?
Oh, and your diagram is delicious. That has nothing to do with 6-food shadows or with actual eating or with any technical mathematical term. I just think delicious is the right word for it.
I’m glad you like it. I enjoy making the diagrams for the blog.
I hope to make it back to this problem soon, when I will elaborate more on my findings and think more about how to know when I’ve found the largest right-truncatable prime.
Tim, I share your reaction; I don’t feel it about math, but I find many colors absolutely “delicious”!
Nom nom.