Determine all possible triplet(s) (p,q,r) of positive rational numbers such that each of p+q+r, p^-1 + q^-1 + r^-1 and p*q*r is an integer.

(In reply to some obvious ones by Charlie)

 1  1  1
 1  2  2
 2  1  2
 2  2  1
 2  3  6
 2  4  4
 2  6  3
 3  2  6
 3  3  3
 3  6  2
 4  2  4
 4  4  2
 6  2  3
 6  3  2

from 

  10   for A=1 to 20
  20   for B=1 to 20
  25   if gcd(A,B)=1 then
  30   :for C=1 to 20
  40   :for D=1 to 20
  45   :if gcd(C,D)=1 then
  50   :for E=1 to 20
  60   :for F=1 to 20
  65   :if gcd(E,F)=1 then
  70     :P=A//B:Q=C//D:R=E//F
  80     :if P*Q*R=int(P*Q*R) then
  90       :if P+Q+R=int(P+Q+R) then
 100         :if 1//P+1//Q+1//R=int(1//P+1//Q+1//R) then
 110           :print P;Q;R
 115   :endif:endif:endif:endif
 120   :next
 130   :next:endif
 140   :next
 150   :next:endif
 160   :next
 170   :next

This covers all rational numbers with no integer above 20 in the numerator or the denominator.  Only integers were found for p, q and r.

Posted by Charlie on 2009-06-16 12:46:42