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 Multiply Together, Get Cubic (Posted on 2007-06-21)
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.

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 re: if s = 1 (one more solution!) | Comment 7 of 8 |
(In reply to if s = 1 (one more solution!) by Steve Herman)

In addition to (-1,0), there's also (-1,-1) where s=1 also, as -1+1+1 = (-1)(-1)(1).

`10   for Tot=0 to 100000020   for P=0 to Tot30   Q=Tot-P40   S=gcd(P,Q)50   if P+Q*Q+S*S*S=P*Q*S then print P;Q51   if -P+Q*Q+S*S*S=-P*Q*S then print -P;Q52   if P+Q*Q+S*S*S=-P*Q*S then print P;-Q53   if -P+Q*Q+S*S*S=P*Q*S then print -P;-Q60   next70   next`
`shows `
` 0  0 0  0 0  0 0  0-1  0-1  0-1 -1 4  2 5  2 5  3 4  6`

but the (0,0) lines are extraneous, as gcd(0,0) is really undefined, but the programming language returns a zero.

Also, as before, the totals were not allowed to get up to 1,000,000, so the upper limit of testing was not tested that high.

Edited on June 22, 2007, 9:37 am
 Posted by Charlie on 2007-06-22 09:32:49

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