Let a, b, c be positive integers such that a, b, c, a+b-c, a+c-b, b+c-a, a+b+c are 7 distinct primes.

The sum of two of a, b, c is 800.

If d be the difference of the largest prime and the least prime among those 7 primes, find the maximum value of d.

   10   for CNo=1 to 139
   20     C=prm(CNo)
   30     P800=C+800
   40     if prmdiv(P800)=P800 then
   50         :M800=800-C
   60         :if prmdiv(M800)=M800 then
   70            :print C,M800,P800
   80   next
  100
  110   for ANo=1 to 139
  120     A=prm(ANo)
  130     B=800-A
  140     if prmdiv(B)=B then
  150         :if A<B and B<C then
  160            :Bc=B+C:Ac=A+C
  170            :if prmdiv(Bc-A)=Bc-A and prmdiv(Ac-B)=Ac-B then:print A,B,C:print " ",A+B-C,Bc-A,Ac-B,A+B+C
  190   next

finds

 797     3       1597
 
 13      787     797
         3       1571    23      1597

 43      757     797
         3       1511    83      1597

 157     643     797
         3       1283    311     1597

 223     577     797
         3       1151    443     1597

The first line means that the only prime which is still prime when 800 is added to it (as in a+b+c) or is itself subtracted from 800 (as in a+b-c or one of the other differences; I chose c), is 797, so c is 797 and a+b+c = 1597.

The remaining four sets are sets that work in order on two lines:

a  b  c
  a+b-c  b+c-a  a+c-b   a+b+c
 
The maximum difference among these is 1597 - 3 = 1594 = d. 

Posted by Charlie on 2013-04-26 17:47:32