The Prime Game

  

Ulam Prime Spiral

150x150 spiral with prime "ley lines"

I read a post some time ago called "Math Notes" by Greg Ross at Futility Closet, and as always, it was very interesting.

One part in particular got my mind turning. He wrote:

1

19

197

1979

19793

197933

1979339

19793393 and

197933933 are all prime.

(OK, we don't use 1 as prime anymore, but it was common a century ago)    

It struck me that this could be a great student game/project.

Such numbers are called right truncatable primes, since you can continuously chop off the last digit  and still have a prime.  There are similar types for  left truncatable, and even some left/right truncatable.  

Player one picks a number, a single prime digit.

Player two must pick a second digit so that the two form a two digit prime.

Play continues until one can not make a prime.

An interesting alternative rule cold be to allow the subsequent numbers to be added at either the front or the end of the string.

It might also be interesting to create a "map" of the strings possible..

For example the one above would be on a mapping that starts with one and then branches to 11, 13, 17, and 19 The 11 could go to 113 but then would be a dead end as 1131, 1133, 1135, and 1137 (and 1139) are all factorable. The 13 node can be extended to 131, 137, 139.

Students might amaze themselves with the long strings they could create.

For the string that started all this above, one might add 1979339333 is also prime, but if you wished, you could use 1979339339 which is prime as well.  That's a quick ten digit string of primes.  It is not easy for students to test if either of those is the end of the string, but along the way they have learned something about testing primes.

A good solitaire version might be to start with the digits 1,2,3,4,5,6,7,8,9 and try to build a tree that leads to a ten digit prime with all the numerals by adding one digit at a time.

And a footnote, Greg had this ten digit prime which isn't truncatable, but has the digits 1 through 9 in order...1234567891.  

Thought people might wonder about left truncatable primes, so here are the three  largest found so far by 

The Prime Glossary page : The three largest left-truncatable primes are:

 959 18918 99765 33196 93967,

 966 86312 64621 65676 29137, and

3576 86312 64621 65676 29137.

If you want to learn more, here is a paper that might be of interest  

C. Caldwell, "Truncatable primes," J. Recreational Math., 19:1 (1987) 30--33. [A recreational note discussing left truncatable primes, right truncatable primes, and deletable primes.]