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Topic: Absolute value question (Read 1333 times) |
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aicoped
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Absolute value question
« on: Feb 13th, 2011, 4:18pm » |
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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.
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towr
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Re: Absolute value question
« Reply #1 on: Feb 14th, 2011, 11:33am » |
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It's a type of Diophantine equation
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 . There's a finite number of solutions for all integer n, and for 6 in particular it has none.
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| « Last Edit: Feb 14th, 2011, 11:34am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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aicoped
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Re: Absolute value question
« Reply #2 on: Feb 14th, 2011, 12:50pm » |
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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.
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JohanC
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Re: Absolute value question
« Reply #3 on: Feb 14th, 2011, 1:45pm » |
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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/ .
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