March 7th, 2013, 06:14 AM  #1 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Diophantine equation
It seems like has all the integer solutions either and/or assumed x < y. Can one give a proof/counterexample? 
March 7th, 2013, 08:14 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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.

March 7th, 2013, 09:48 AM  #3  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Diophantine equation
Okay, thanks. I was being fooled by the behavior of small d's. Quote:
 
March 7th, 2013, 10:51 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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.

March 7th, 2013, 11:47 AM  #5  
Senior Member Joined: Sep 2010 Posts: 221 Thanks: 20  Re: Diophantine equation Quote:
Let x=ky where k  rational number; dky^2(ky)^3=y^3 or dk=y(k^3+1); k(dyk^2)=y; If d=2y then k=1; d=(9/2)y then k=2; d=(28/3)y then k=3; etc.  
March 7th, 2013, 12:18 PM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  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.

March 7th, 2013, 04:54 PM  #7 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT  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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