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 Rationally Integers (Posted on 2009-06-16)
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.

 No Solution Yet Submitted by K Sengupta No Rating

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 some more via computer program Comment 2 of 2 |
(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

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