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        riddles >> medium >> Absolute value question
        
(Message started by: aicoped on Feb 13^th, 2011, 4:18pm)

Title: Absolute value question
Post by aicoped on Feb 13^th, 2011, 4:18pm

Given the absolute values of x^3-y^2=6 and assuming x and y must be positive integers, are there any solutions to this equation, or more generally x^3-y^2=z. solve for all positive integer z.

z=1 x=2 y=3
z=2 x=3 y=5

and so on.

I did a spread sheet for a pretty large range of values and it did not yield several answers(6 being among those). I recall reading somewhere that there is no proof that every answer exists or a general method of solving for all z. if someoen could redirect me to that result, I would appreciate that as well. Also, if anyone can solve for 6, I would like to see the answer.

Title: Re: Absolute value question
Post by towr on Feb 14^th, 2011, 11:33am

It's a type of Diophantine equation (http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html)
Ones of the form y^2 = x^3 + n apparently gives an elliptic curve (I really should read up on those some day) known as the Mordell curve (http://mathworld.wolfram.com/MordellCurve.html). There's a finite number of solutions for all integer n, and for 6 in particular it has none.

Title: Re: Absolute value question
Post by aicoped on Feb 14^th, 2011, 12:50pm

OK thanks towr. OK now what is the proof that those numbers dont have solutions? Is there a formula that derives the ones that don't?

Is there a way to plug in a number and yield that it has certain solutions or is that one of the unprovable things about those forms of equations.

thanks for the help. in advance.

Title: Re: Absolute value question
Post by JohanC on Feb 14^th, 2011, 1:45pm

Checking out the online links given in the MathWorld article referenced by Towr seems to indicate that it is very well understood problem. Given enough mathematical background, one could calculate the exact number of solutions for a given n. But the references also seem to be unaware of "elementary proofs".

Note that you have to both consider n=-6 as n=6 to get to your original equation (for which there are no solutions).
All the solutions for n going from -100000 to +100000 are given at http://tnt.math.se.tmu.ac.jp/simath/MORDELL/ (http://tnt.math.se.tmu.ac.jp/simath/MORDELL/).