Starting with a prime digit add another digit, after or before the first, then another, adding it after, before or within the second number, and continue, keeping the resulting numbers prime, without repeating any of the digits you have used so far.
Example: 2, 23, 263, 2063, 29063....
Obviously you cannot reach a pandigital number (it will always be divisible by 9)
.
What are the lowest and highest numbers in the set of eligible solutions with the maximum number of digits?
Please specify the interim stages leading to your results.
I found a neat javascript utility to assist with this:
http://javascript.internet.com/mathrelated/primenumber.html. It could be a utility worthy of installing on your own computer as a web page for your browser to call via Bookmarks/Favorites, or just bookmark the site.
My first challenge is get a prime which is a close to
123456789 but that number is divisible by 3.
Now if I substitute the "9" for a "0" my lowest target is
102345678, but that is also divided by 3.
Playing with the 10 digits in increasing order, but rearranging the "1" and the "0" the lowest available prime with 9 digits is
102345689.
The removal of any digit between the "1" and the "9" in that number results in a composite so that is an unreachable target.
The number 102345697 is the next smaller available prime with 9 digits and it is reachable under the rules as defined.
7
17
107
1097
10597
104597
1034597
10345697
102345697 ****** Lowest
From the other end,
976854301 seems to be the largest available from the 10 digits as defined.
Beginning with 3 shows promise:
3
31
431
5431
75431
754301
8754301
98754301
The problem now is that the inclusion of a "2" or a "3" means that there divisibility by 3.
Beginning with a 5 only gets up to 3 digits following my plan to reach my target.
I have success by beginning with a "7".
7
97
971
9721
97021
970421
9708421
97068421
970638421 ******
I believe the Highest and Lowest, according to the define procedure, are
970638421 and
102345697.

Posted by brianjn
on 20110304 19:56:50 