perplexus.info :: Just Math : Integer Pair and Infinite Puzzle

Integer Pair and Infinite Puzzle (Posted on 2015-09-20)

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

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

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

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (0)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info