Thread: A back and forth prime type of problem

To illustrate what I have in mind, let's start with 137.

137 is a prime number. Take off the 1 leaves you with 37 which is another prime number and taking off the seven leaves you with 3 which is yet another prime.

By reversing what I did, you start with 3, then add in the 7 digit on the right-hand side then add in the one on the left-hand side, all prime numbers. Now maybe there's a digit you can add on the right-hand side of 137 say 137a, yielding another prime number. And then another digit you can add to the left-hand side yielding b137a which (hopefully) is yet another prime number.

The puzzle is to find the string of prime numbers, growing in the manner just illustrated, growing to the largest prime number that can be determined by the pattern I just illustrated (you don't have to start with 137, you have the choice of digits 1-9 on the left-hand side to grow with and the digits 1,3,5,7 and 9 to grow with on the right-hand side).

How far can you get?

There's 55 that are 9digits long, lowest : 122479913, highest : 992859779

7,79,479,4799,24799,247991,2247991,22479913,122479 913 : that follows the rules, right?

Will try for longer later...

I found 25 with 11 digits: low = 22212713317, high = 99291733379

There are several ways to vary this puzzle. Here are two examples:

Using 3 e.g., you can start off adding a single digit to the left- or right-hand side, then add a two-digit number to the other side, then a three-digit number to the other side again, then a four-digit number to the other side again until you're forced to stop when all you get is a composite number.

Another variation is to have a3b where you let a = x^2 and b = y^2 to (hopefully) get two more prime numbers, then you would have ca3bd where c = s^2 and d = t^2 which would generate two more prime numbers and then see how far you can take it.

Such a puzzle I would call a superpuzzle due to the variations (plus there's more I've thought of).

Have fun.