Determine all possible integer pairs (p,q) such that p+q²+s³=pqs, where s=gcd(p,q) and gcd denotes the greatest common divisor.

(In reply to early results -- no proof by Charlie)

For positive and negative p and q up to abs(p)+abs(q) = 38,000, the same set of four were the only ones found. 

10   for Tot=3 to 1000000
20   for P=1 to Tot-1
30   Q=Tot-P
40   S=gcd(P,Q)
50   if P+Q*Q+S*S*S=P*Q*S then print P;Q
51   if -P+Q*Q+S*S*S=-P*Q*S then print -P;Q
52   if P+Q*Q+S*S*S=-P*Q*S then print P;-Q
53   if -P+Q*Q+S*S*S=P*Q*S then print -P;-Q
60   next
70   next

(the gcd always is considered positive)

Posted by Charlie on 2007-06-21 11:05:34