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Integer Pair and Infinite Puzzle (Posted on 2015-09-20) Difficulty: 3 of 5
Each of p and q is a positive integer with gcd (p,q) =1 and the equation:
(x+p)3 = q*x has exactly three distinct integer solutions in x.

Do there exist infinitely many pairs (p,q) satisfying the given conditions? Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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I've got your cubes here, jer Comment 2 of 2 |
Multiply out (x-a)(y-b)(x-c) and equate coefficients with the given equation.

3p=-(a+b+c)
3p^2-q=ab+ac+bc
p^3=-abc

Setting (a,b,c) to be cubes is an easy next step but solving the first equation is much harder.  I also tried (1,p,p^2) with no better success.

Using your solution gives p=30 and q=29791 which equals, get ready, 31^3. 

  Posted by xdog on 2015-09-23 19:42:01
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