Problems & Puzzles: Puzzles

Puzzle 157.  Zip primes

Walter Schneider posts the following puzzle:

Let us look at primes which remain prime when "zipping" the number into k parts by alternately distributing the left-most digit to the parts.

For the number 27239 for example the zipping goes as follows:

k=1:     27239    prime

k=2:     229      prime

         73       prime 

k=3:     23       prime 

         79       prime 

         2        prime

k=4:     29       prime 

         7        prime

         2        prime

         3        prime

k=5:     2        prime

         7        prime

         2        prime

         3        prime

         9        not prime

We call a n-digit number N k-zippable (k=1,...,n) if all k parts are prime. If this is true for k=1,2,...,K we call N a zip prime of order K. If K is maximal (i.e. K=n) we say N is a perfect zip prime.

The example above shows that 27239 is a zip prime of order 4.

Questions:

1. Show that there are only a finite quantity of perfect zip primes and obtain all of them.

 2. Find five zip primes of order 7.

 3. Find a zip prime of order 8, 9 and 10.

 


Solution:

Sudipta Das sent (October 29, 2002) the following answers to this puzzle. After this we only need a zip prime of order 9 & 10

1. The perfect zip primes are :
 
Order 1 : 2 , 3 , 5 , 7
Order 2 : 23 , 37 , 53 , 73
Order 3 : 223 , 233 , 337 , 523 , 733 , 773
Order 4 : 5233
Order 5 : 33377 , 72733
Order 6 : 272333 , 572333
Order 7 : 5222333
 
The primes obtained for each k ( k = 1 , 2 , ... , order - 1 ) are zip primes themselves .
 
2. The first 5 zip primes of Order 7 are :
5222333 , 1112777999 , 244712331139 , 41311151703973937 , 100070904451773119
 
3. The first zip prime of Order 8 is 82845223933399

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