Combinations of numbers in squares.

by individ, Dec 9, 2015, 12:08 PM

$$ \left\{\begin{aligned}&a+b+N=x^2\\&a+c+N=y^2\\&b+c+N=z^2\\&a+b+c+N=w^2\end{aligned}\right. $$
$$ a=-t(2t^2+3t+k^2+1-N) $$
$$ b=(t+1)(2t^2+t+k^2-N) $$
$$c=(t^2+t+\frac{(k-1)^2-N}{2})(t^2+t+\frac{(k+1)^2-N}{2}) $$
$$ x=k $$
$$y=t^2+\frac{k^2-1-N}{2} $$
$$ z=t^2+2t+\frac{k^2+1-N}{2} $$
$$ w=t^2+t+\frac{k^2+1-N}{2} $$
More interesting the other decisions - when the number is positive.

$$a=(3q+p+3s+1)(410q^2+140qp+500qs+170s^2+100ps+243q+41p+147s+15p^2+36-5N)$$
$$b=(41q^2+14qp+50qs+17s^2+10ps+27q+4p+15s+\frac{3p^2+9-N}{2})(369q^2+126qp+450qs+153s^2+90ps+219q+38p+135s+\frac{27p^2+65-9N}{2})$$
$$c=(82q^2+28qp+100qs+34s^2+20ps+41q+7p+25s+3p^2+5-N)(328q^2+112qp+400qs+136s^2+80ps+205q+35p+125s+12p^2+32-4N)$$
$$x=123q^2+42qp+150qs+51s^2+30ps+82q+14p+50s+\frac{9p^2+27-3N}{2}$$
$$y=164q^2+56qp+200qs+68s^2+40ps+96q+17p+60s+2(3p^2+7-N)$$
$$z=205q^2+70qp+250qs+85s^2+50ps+120q+20p+72s+\frac{5(3p^2+7-N)}{2}$$
$$w=205q^2+70qp+250qs+85s^2+50ps+123q+21p+75s+\frac{15p^2+37-5N}{2}$$
$ N - $ the number is specified by the problem statement and can be any.
This post has been edited 1 time. Last edited by individ, Dec 9, 2015, 12:15 PM
number theory
Diophantine equation
system of equations

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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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