A system of similar nonlinear equations

by individ, Dec 29, 2022, 12:45 PM

There is a whole group of such systems of equations.

$$ \left\{\!\begin{aligned} & A^2+AB+B^2=C^2 \\ & F^2+FZ+Z^2=C^2 \end{aligned}\right. $$
Such a system is solved as standard. First, we write down the parametrization of one equation.

$$A=p^2-s^2$$
$$B=s(s+2p)$$
$$C=p^2+ps+s^2=x^2+xy+y^2$$
$$F=x^2-y^2$$
$$Z=y(y+2x)$$
And then we find the parameterization for the necessary parameters.

$$p=2q^2+2qt-t^2-3qk+k^2$$
$$s=-q^2+2qt+2t^2-3tk+k^2$$
$$y=q^2+qt+t^2-k^2$$
$$x=q^2+qt+t^2-3(q+t)k+2k^2$$
Diophantine equation
system of equations
number theory

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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
number theory
system of equations
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