Let capitalised letters of the alphabet stand for 4digit prime numbers each of whose digits is distinct and also prime, so that e.g. A=5237.
A pair of such letters stands for the concatenation of two such numbers to form an 8digit number, e.g. AA=52375237
Denote 8 different such 4digit primes by the letters of the word CAPTURES, so that each of CP,ET,PU,RU,TE,TR, and UC is also prime.
Given that A=5237, what is the value of S?
When this puzzle was in the queue, I hadn't thought of the limitations on the 4digit primes, and was given a solution by the poser, broll, but only perused is far enough to see that there were only 8 candidate 4digit primes. I didn't go through the remainder of the solution. But now that I've solved it myself, I see there are two solutions.
First the program and its findings:
5 dim Pr4(10)
10 N$="7532":H$=N$:PCt=0
15 repeat
20 gosub *Permute(&N$)
30 if prmdiv(val(N$))=val(N$) then inc PCt:print PCt;N$:Pr4(PCt)=val(N$)
35 until N$=H$
100 for A=1 to PCt:for B=1 to PCt
110 Pr8=10000*Pr4(A)+Pr4(B)
120 if prmdiv(Pr8)=Pr8 then print Pr8;A;B
199 next B:next A
740 end
800
The permute algorithm is listed elsewhere on the site.
1 2357
2 2753
3 3257
4 3527
5 5237
6 5273
7 7253
8 7523
23572753 1 2
23573527 1 4
23577523 1 8
27533257 2 3
32575273 3 6
35272357 4 1
52732357 6 1
52737253 6 7
72533257 7 3
72537523 7 8
75235273 8 6
First the 4digit primes are numbered 1 to 8. Then the formable 8digit primes are listed, with the numbers of the two 4digit primes that make up each one.
Two possible sets of assignment work. I show the ordinal number of the 4digit prime beneath the corresponding letter:
CAPTURES CAPTURES
65713248 75316842
In the first set, S equates to the 8th 4digit prime, 7523. In the second set, S equates to the 2nd 4digit prime, 2753.

Posted by Charlie
on 20110614 14:46:24 