I have been trying to come up with my own math problems recently and this is one of my first. It introduces the idea of an extreme prime. I hope that an extreme prime isn't already a thing, because I just used the name to describe a special number. I have a solution to the problem, but I'd like to see smarter solutions and get some feedback on the problem so I can make better ones in the future.
An extreme prime is a number such that every number within the number is prime, expect one-digit numbers, and the number itself is prime. Examples are below for clarity, as I'm bad at explaining.
Examples:
617 is a prime
61 is a prime, and 17 is a prime. Therefore 617 is an extreme prime. Note 6 is composite: the digits need not be prime.
1373 is prime.
13 is prime, 37 is prime, 73 is prime, 137 is prime, 373 is prime. Therefore 1317 is an extreme prime. Fun fact: 373 is also the only 3 digits extreme prime where the digits are prime, so I guess it must be ultra-prime :) :)
$617$ is a prime. Also, $61$ is a prime and $17$ is a prime. Therefore $617$ is an extreme prime. Note $6$ is composite: the digits need not be prime.
$1373$ is prime. Also, $13$ is prime, $37$ is prime, $73$ is prime, $137$ is prime, $373$ is prime. Therefore $1317$ is an extreme prime. Fun fact: $373$ is also the only $3$ digits extreme prime where the digits are prime, so I guess it must be ultra-prime.
The question is to prove that no 5$5$ digit extreme prime exists. I'm looking forward to some feedback and some ways I can word what an extreme prime is, hope it is fun to solve :).
Some other facts I noticed when checking my proof with python (which I do not have a proof for): you may like to try to prove them.
A 3 digit extreme prime cannot contain a 2,8 or 5.
a 4 digit extreme prime cannot contain a 2,8,5 or 4
a 4 digit extreme prime never starts with 7
A $3$ digit extreme prime cannot contain a $2,8$ or $5$.
A $4$ digit extreme prime cannot contain a $2, 8, 5$ or $4$.
A $4$ digit extreme prime never starts with $7$.
Quite a few super primes (primes that occupy prime numbered positions in the sequence of all prime numbers) are extreme primes. Can you find them all and create the prime-st number set of all time! :)
