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March 7th, 2013, 05:14 AM   #1
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Diophantine equation

It seems like



has all the integer solutions either and/or assumed x < y.

Can one give a proof/counterexample?
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March 7th, 2013, 07:14 AM   #2
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Re: Diophantine equation

(x, y, d) = (6, 18, 56). There are infinitely many such counterexamples; probably the counterexamples are asymptotically dominant with any reasonable measure.
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March 7th, 2013, 08:48 AM   #3
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Re: Diophantine equation

Okay, thanks. I was being fooled by the behavior of small d's.

Quote:
Originally Posted by CRGreathouse
probably the counterexamples are asymptotically dominant with any reasonable measure.
Can you elaborate a bit more?
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March 7th, 2013, 09:51 AM   #4
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Re: Diophantine equation

If you find all (x, y, d) satisfying 0 < x <= y < M and the equation, I guess that the percentage which meet your conditions have asymptotic density 0.
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March 7th, 2013, 10:47 AM   #5
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Re: Diophantine equation

Quote:
Originally Posted by mathbalarka
Can you elaborate a bit more?
dxy-x^3=y^3;
Let x=ky where k - rational number;
dky^2-(ky)^3=y^3 or dk=y(k^3+1);
k(d-yk^2)=y;
If d=2y then k=1;
d=(9/2)y then k=2;
d=(28/3)y then k=3; etc.
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March 7th, 2013, 11:18 AM   #6
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Re: Diophantine equation

The only thing that matters in this problem is that y | x^3 and x | y^3, so just make sure that they have the same primes and they're within a multiple of 3 of each other. So if one is divisible by (exactly) 2^4, the other needs to have 2 raised to a power between 2 = ceil(4/3) and 12 = 4*3.
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March 7th, 2013, 03:54 PM   #7
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Re: Diophantine equation

I'm not sure if this will be helpful, but your equation can be written in parametric form as





For the derivation of these equations see, for example, the Wikipedia article on the Folium of Descartes.

In particular, y = tx. In CRG's solution (6, 18, 56), we have that t = 3.
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