Prove that a+b+c+d is a composite number

by individ, Oct 30, 2022, 8:33 AM

[url]http://math.hashcode.ru/questions/249507/теория-чисел-доказать-что-a-b-c-d-составное-число?страница=1&focusedAnswerId=249550#249550[/url]

$$X^2+aXY+Y^2=Z^2+aZV+V^2$$
Решений можно записать два. Одно когда взаимно простые....

$$X=(a+1)k^2+2tk-t^2-(a+2)qk+q^2$$
$$Y=(a+1)t^2+2tk-k^2-(a+2)qt+q^2$$
$$Z=k^2+akt+t^2-q^2$$
$$V=k^2+akt+t^2+(a+1)q^2-(a+2)(t+k)q$$
Сумма их имеет вид.

$$X+Y+Z+V=(a+2)(t+k-q)^2$$
И когда нет...

$$X=k^2+akt+t^2-q^2$$
$$Y=q(aq-at-2k)$$
$$Z=t(aq-at-2k)$$
$$V=t^2-k^2-q^2+aqk$$
$$X+Y+Z+V=(a-2)(q+t)(q+k-t)$$
Ну и дальше всё совсем просто... взаимно простые когда скобки некоторые равны 1... ну или -1

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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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