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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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 If s = 2 | Comment 5 of 8 |
If s = 2,    p + q^2 + 8 = 2pq

p = (q^2 + 8)/(2q - 1)

4p = (4q^2 + 32)/(2q - 1) = (4Q^2 - 1 + 33)/(2q - 1) =
(2q + 1)  + (33/(2q - 1))

A necessary (but not sufficient) condition for p integral is that 4p is integral, and this can only happen if (2q - 1) is 3 or 11 or 1 or -1 or -3 or -11.  i.e, if q = 2 or 6 or 1 or 0 or -1 or -5.  But if s = 2, only 2 or 6 are possible values of q, because 2 cannot be a gcd of 1, 0 , -1 , or -5.  Both q = 2 and 6 lead to solutions:

(p,q) = (4,2) and (4,6)

Next step: s > 2

 Posted by Steve Herman on 2007-06-21 19:36:00

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