Continuation .

by individ, Aug 2, 2014, 10:51 AM

You can write another solution. This is equivalent to the equation:

\[x^2+xy+y^2=z^3\]

The solutions have the form:

\[x=s^3+3ps^2-p^3\]

\[y=p^3+3sp^2-s^3\]

\[z=p^2+ps+s^2\]

\[...\]

\[x=s(p^2+ps+s^2)\]

\[y=p(p^2+ps+s^2)\]

\[z=p^2+ps+s^2\]

\[...\]

Will make a replacement.

\[b=3p^2+6ps+2s^2\]

\[t=6s^2+6ps\]

\[q=3p^2+6ps+4s^2\]

The solution has the form:

\[x=q(3b^2-6bt-t^2)\]

\[y=q(3b^2+6bt-t^2)\]

\[z=3q^2\]
number theory
Diophantine equation

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  • How did you discover these parametric solutions to diophantine equations?

    by fanzhuyifan, Dec 31, 2016, 9:25 AM

  • Russian? are you sure it ain't greek?

    by Mathisfun04, Dec 27, 2016, 4:03 PM

  • yep i agree

    by Eugenis, Oct 31, 2015, 2:40 AM

  • Best blog ever

    by FlakeLCR, Oct 13, 2015, 8:07 PM

  • too much russian.

    by rileywkong, Aug 21, 2015, 6:10 PM

  • Decided the equation.

    by individ, Aug 20, 2015, 5:05 AM

  • Some insight into how you figured it out?

    by Not_a_Username, Aug 19, 2015, 3:52 PM

  • I figured it out. Decided equation.

    by individ, Aug 19, 2015, 5:01 AM

  • Yes, how do you come up with the formula? :P

    by Not_a_Username, Aug 18, 2015, 10:29 PM

  • I don't understand. There are the equation and there is a formula to it solutions. What is the problem?

    by individ, Aug 13, 2015, 4:22 PM

  • What? Lol you are substituting solutions with literally no motivation

    by Not_a_Username, Aug 13, 2015, 12:59 PM

  • What replacement? Where?

    by individ, Aug 8, 2015, 5:37 AM

  • Darn, what are the motivation for these substitutions???

    by Not_a_Username, Aug 5, 2015, 10:44 AM

  • Are you greek?

    by beanielove2, Dec 24, 2014, 6:31 PM

  • So, a purely mathematical blog?

    by Lionfish, Dec 2, 2014, 1:20 PM

  • To prove that it is necessary to show the method of calculation. I do not want to do yet.

    by individ, Mar 28, 2014, 6:14 AM

  • I can't understand these posts....What language are they written in? I don't recognize it.

    I like your avatar! :P

    by 15cjames, Mar 11, 2014, 1:57 PM

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Diophantine equation
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