Diophantine equation 3-rd degree.
by individ, Jan 22, 2015, 8:14 AM
When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows.
![\[b(x^3+y^3)=az^3\]](//latex.artofproblemsolving.com/d/2/7/d27c9788131773f7e030dccf6d5e406e3ca22932.png)
Where
![\[b=q^2+3n^2\]](//latex.artofproblemsolving.com/b/9/2/b9224a626924d216cafe498ed19b2805887fd136.png)
![\[a=2(q^2-3n^2)\]](//latex.artofproblemsolving.com/0/4/8/048f5ee80b0e36fc36ed4e086178bcbbb2b4ec3f.png)
When you can represent the coefficients in this form. Where
any number.
For simplicity we make a replacement.
![\[t=2qs\]](//latex.artofproblemsolving.com/8/9/a/89adba5d6037f07138c331fd8c725b3a818c4634.png)
![\[p=2qj\]](//latex.artofproblemsolving.com/5/7/4/574c69091386869da17ed5be902bf3cb8c3f847d.png)
![\[k=(q+3n)s+(q-3n)j\]](//latex.artofproblemsolving.com/6/4/d/64d8209a9e9480eaf93eb7cbf6ae0694cbb7d8a4.png)
- integers asked us.
Then the solution can be written in this form.
![\[x=p^2-k^2+2kt-2pt\]](//latex.artofproblemsolving.com/6/f/b/6fb2be61caeb99d310f2c84410097cd56d385045.png)
![\[y=t^2-k^2+2kp-2pt\]](//latex.artofproblemsolving.com/d/4/7/d4799d50ee64ddd5040be11e075afe0d5ce187f2.png)
![\[z=p^2+k^2+t^2-pt-pk-tk\]](//latex.artofproblemsolving.com/b/5/5/b5506c71aedb1a1a245439886bc3a5b2b5a7ca2b.png)
I was already glad that there are such factors when their values have infinitely many solutions. But it turned out that it turns out one mutually simple solution. All other solutions are multiples of him.
![\[b(x^3+y^3)=az^3\]](http://latex.artofproblemsolving.com/d/2/7/d27c9788131773f7e030dccf6d5e406e3ca22932.png)
Where
![\[b=q^2+3n^2\]](http://latex.artofproblemsolving.com/b/9/2/b9224a626924d216cafe498ed19b2805887fd136.png)
![\[a=2(q^2-3n^2)\]](http://latex.artofproblemsolving.com/0/4/8/048f5ee80b0e36fc36ed4e086178bcbbb2b4ec3f.png)
When you can represent the coefficients in this form. Where
any number.For simplicity we make a replacement.
![\[t=2qs\]](http://latex.artofproblemsolving.com/8/9/a/89adba5d6037f07138c331fd8c725b3a818c4634.png)
![\[p=2qj\]](http://latex.artofproblemsolving.com/5/7/4/574c69091386869da17ed5be902bf3cb8c3f847d.png)
![\[k=(q+3n)s+(q-3n)j\]](http://latex.artofproblemsolving.com/6/4/d/64d8209a9e9480eaf93eb7cbf6ae0694cbb7d8a4.png)
- integers asked us.Then the solution can be written in this form.
![\[x=p^2-k^2+2kt-2pt\]](http://latex.artofproblemsolving.com/6/f/b/6fb2be61caeb99d310f2c84410097cd56d385045.png)
![\[y=t^2-k^2+2kp-2pt\]](http://latex.artofproblemsolving.com/d/4/7/d4799d50ee64ddd5040be11e075afe0d5ce187f2.png)
![\[z=p^2+k^2+t^2-pt-pk-tk\]](http://latex.artofproblemsolving.com/b/5/5/b5506c71aedb1a1a245439886bc3a5b2b5a7ca2b.png)
I was already glad that there are such factors when their values have infinitely many solutions. But it turned out that it turns out one mutually simple solution. All other solutions are multiples of him.
June 2025
December 2022