Задача о четырёх кубах

Еще Платону было известно, что сумма кубов сторон пифагорейского треугольника также является кубом ${\displaystyle 3^{3}+4^{3}+5^{3}=6^{3}}$^[2], о чем он упоминает в своем «Государстве»^[3].

Наименьшие натуральные решения:

${\displaystyle 3^{3}+4^{3}+5^{3}=6^{3}}$

${\displaystyle 1^{3}+6^{3}+8^{3}=9^{3}}$

${\displaystyle 3^{3}+10^{3}+18^{3}=19^{3}}$

${\displaystyle 7^{3}+14^{3}+17^{3}=20^{3}}$

${\displaystyle 4^{3}+17^{3}+22^{3}=25^{3}}$

${\displaystyle 18^{3}+19^{3}+21^{3}=28^{3}}$

${\displaystyle 11^{3}+15^{3}+27^{3}=29^{3}}$

${\displaystyle 2^{3}+17^{3}+40^{3}=41^{3}}$

${\displaystyle 6^{3}+32^{3}+33^{3}=41^{3}}$

${\displaystyle 16^{3}+23^{3}+41^{3}=44^{3}}$

Если разрешить отрицательные значения, то имеют место тождества:

${\displaystyle -1^{3}+9^{3}+10^{3}=12^{3}}$

${\displaystyle -2^{3}+9^{3}+15^{3}=16^{3}}$

${\displaystyle -2^{3}+15^{3}+33^{3}=34^{3}}$

${\displaystyle -2^{3}+41^{3}+86^{3}=89^{3}}$

${\displaystyle -3^{3}+22^{3}+59^{3}=60^{3}}$

Г. Харди и Райт (1938)^[4][5]

• ${\displaystyle x=-a(b-3c)(b^{2}+3c^{2})+a^{4}}$

  ${\displaystyle y=\quad a(b+3c)(b^{2}+3c^{2})-a^{4}}$

  ${\displaystyle z=\quad a^{3}(b-3c)-(b^{2}+3c^{2})^{2}}$

  ${\displaystyle w=\quad a^{3}(b+3c)-(b^{2}+3c^{2})^{2}}$

Н. Элкис^[1]

${\displaystyle {\begin{cases}x=d(-(s+r)t^{2}+(s^{2}+2r^{2})t-s^{3}+rs^{2}-2r^{2}s-r^{3}),\\y=d(t^{3}-(s+r)t^{2}+(s^{2}+2r^{2})t+rs^{2}-2r^{2}s+r^{3}),\\z=d(-t^{3}+(s+r)t^{2}-(s^{2}+2r^{2})t+2rs^{2}-r^{2}s+2r^{3}),\\w=d((s-2r)t^{2}+(r^{2}-s^{2})t+s^{3}-rs^{2}+2r^{2}s-2r^{3})\end{cases}}}$

Леонард Эйлер, 1740 год

• ${\displaystyle x=1-(a-3b)(a^{2}+3b^{2})}$

  ${\displaystyle y=-1+(a+3b)(a^{2}+3b^{2})}$

  ${\displaystyle z=-a-3b+(a^{2}+3b^{2})^{2}}$

  ${\displaystyle w=-a+3b+(a^{2}+3b^{2})^{2}}$

Линник, 1940 год

• ${\displaystyle x=b(a^{6}-b^{6})}$

  ${\displaystyle y=a(a^{6}-b^{6})}$

  ${\displaystyle z=b(2a^{6}+3a^{3}b^{3}+b^{6})}$

  ${\displaystyle w=a(a^{6}+3a^{3}b^{3}+2b^{6})}$

• ${\displaystyle x=a^{2}(b^{6}-7)+9ac-3c^{2}}$

  ${\displaystyle y=a^{2}{\big [}b^{3}(2b^{3}+9)+7{\big ]}-3ac(2b^{3}+3)+3c^{2}}$

  ${\displaystyle z=a^{2}b{\big [}b^{3}(b^{3}+3)+2{\big ]}-3abc(b^{3}+2)+3bc^{2}}$

  ${\displaystyle w=a^{2}b{\big [}b^{3}(b^{3}+6)+11{\big ]}-3abc(b^{3}+4)+3bc^{2}}$

• ${\displaystyle x=3a^{2}(b^{6}-7)-9ac-c^{2}}$

  ${\displaystyle y=3a^{2}{\big [}b^{3}(2b^{3}-9)+7{\big ]}-3ac(2b^{3}-3)+c^{2}}$

  ${\displaystyle z=3a^{2}b{\big [}b^{3}(b^{3}-6)+11{\big ]}-3abc(b^{3}-4)+bc^{2}}$

  ${\displaystyle w=3a^{2}b{\big [}b^{3}(b^{3}-3)+2{\big ]}-3abc(b^{3}-2)+bc^{2}}$

Роджер Хит-Браун, 1993 год^[6]

• ${\displaystyle x=9a^{4}}$

  ${\displaystyle y=3a-9a^{4}}$

  ${\displaystyle z=1-9a^{3}}$

  ${\displaystyle w=1}$

Морделл, 1956 год

• ${\displaystyle x=9a^{3}b+b^{4}}$

  ${\displaystyle y=9a^{4}}$

  ${\displaystyle z=-b^{4}}$

  ${\displaystyle w=9a^{4}+3ab^{3}}$

• ${\displaystyle x=9a^{3}b-b^{4}}$

  ${\displaystyle y=9a^{4}-3ab^{3}}$

  ${\displaystyle z=b^{4}}$

  ${\displaystyle w=9a^{4}}$

• ${\displaystyle x=9a^{3}b+b^{4}}$

  ${\displaystyle y=9a^{3}b-b^{4}}$

  ${\displaystyle z=9a^{4}-3ab^{3}}$

  ${\displaystyle w=9a^{4}+3ab^{3}}$

Решение, полученное методом алгебраической геометрии

• ${\displaystyle x=3a\left(a^{2}+ab+b^{2}\right)-9}$

  ${\displaystyle y=\left(a^{2}+ab+b^{2}\right)^{2}-9a}$

  ${\displaystyle z=3\left(a^{2}+ab+b^{2}\right)(a+b)+9}$

  ${\displaystyle w=\left(a^{2}+ab+b^{2}\right)^{2}+9(a+b)}$

Рамануджан

• ${\displaystyle x=3a^{2}+5ab-5b^{2}}$

  ${\displaystyle y=4a^{2}-4ab+6b^{2}}$

  ${\displaystyle z=5a^{2}-5ab-3b^{2}}$

  ${\displaystyle w=6a^{2}-4ab+4b^{2}}$

• ${\displaystyle x=a^{7}-3a^{4}(1+b)+a(2+6b+3b^{2})}$

  ${\displaystyle y=2a^{6}-3a^{3}(1+2b)+1+3b+3b^{2}}$

  ${\displaystyle z=a^{6}-1-3b-3b^{2}}$

  ${\displaystyle w=a^{7}-3a^{4}b+a(3b^{2}-1)}$

• ${\displaystyle x=-a^{2}+9ab+b^{2}}$

  ${\displaystyle y=a^{2}+7ab-9b^{2}}$

  ${\displaystyle z=2a^{2}-4ab+12b^{2}}$

  ${\displaystyle w=2a^{2}+10b^{2}}$

Неизвестный автор, 1825 год

• ${\displaystyle x=a^{9}-3^{6}}$

  ${\displaystyle y=-a^{9}+3^{5}a^{3}+3^{6}}$

  ${\displaystyle z=3^{3}a^{6}+3^{5}a^{3}}$

  ${\displaystyle w=3^{2}a^{7}+3^{4}a^{4}+3^{6}a}$

Д. Лемер, 1955 год

• ${\displaystyle x=3888a^{10}-135a^{4}}$

  ${\displaystyle y=-3888a^{10}-1296a^{7}-81a^{4}+3a}$

  ${\displaystyle z=3888a^{9}+648a^{6}-9a^{3}+1}$

  ${\displaystyle w=1}$

В. Б. Лабковский

• ${\displaystyle x=4b^{2}-11b-21}$

  ${\displaystyle y=3b^{2}+11b-28}$

  ${\displaystyle z=5b^{2}-7b+42}$

  ${\displaystyle w=6b^{2}-7b+35}$

Харди и Райт

• ${\displaystyle x=a(a^{3}-2b^{3})}$

  ${\displaystyle y=b(2a^{3}-b^{3})}$

  ${\displaystyle z=b(a^{3}+b^{3})}$

  ${\displaystyle w=a(a^{3}+b^{3})}$

• ${\displaystyle x=a(a^{3}-b^{3})}$

  ${\displaystyle y=b(a^{3}-b^{3})}$

  ${\displaystyle z=b(2a^{3}+b^{3})}$

  ${\displaystyle w=a(a^{3}+2b^{3})}$

Г. Александров, 1972 год

• ${\displaystyle x=7a^{2}+17ab-6b^{2}}$

  ${\displaystyle y=42a^{2}-17ab-b^{2}}$

  ${\displaystyle z=56a^{2}-35ab+9b^{2}}$

  ${\displaystyle w=63a^{2}-35ab+8b^{2}}$

• ${\displaystyle x=7a^{2}+17ab-17b^{2}}$

  ${\displaystyle y=17a^{2}-17ab-7b^{2}}$

  ${\displaystyle z=14a^{2}-20ab+20b^{2}}$

  ${\displaystyle w=20a^{2}-20ab+14b^{2}}$

• ${\displaystyle x=21a^{2}+23ab-19b^{2}}$

  ${\displaystyle y=19a^{2}-23ab-21b^{2}}$

  ${\displaystyle z=18a^{2}+4ab+28b^{2}}$

  ${\displaystyle w=28a^{2}+4ab+18b^{2}}$

• ${\displaystyle x=3a^{2}+41ab-37b^{2}}$

  ${\displaystyle y=37a^{2}-41ab-3b^{2}}$

  ${\displaystyle z=36a^{2}-68ab+46b^{2}}$

  ${\displaystyle w=46a^{2}-68ab+36b^{2}}$

• ${\displaystyle x=-4a^{2}+22ab-9b^{2}}$

  ${\displaystyle y=36a^{2}-22ab+b^{2}}$

  ${\displaystyle z=40a^{2}-40ab+12b^{2}}$

  ${\displaystyle w=48a^{2}-40ab+10b^{2}}$

Аджай Чоудхри, 1998 год^[7]

• ${\displaystyle dx_{1}=(a^{4}+2a^{3}b+3a^{2}b^{2}+2ab^{3}+b^{4})+(2a+b)c^{3},}$

  ${\displaystyle dx_{2}=-\{a^{4}+2a^{3}b+3a^{2}b^{2}+2ab^{3}+b^{4}-(a-b)c^{3}\},}$

  ${\displaystyle dx_{3}=c(-a^{3}+b^{3}+c^{3}),}$

  ${\displaystyle dx_{4}=-\{(2a^{3}+3a^{2}b+3ab^{2}+b^{3})c+c^{4}\},}$

где числа ${\displaystyle a,b,c}$ — произвольные целые, а число ${\displaystyle d\neq 0}$ выбрано таким образом, чтобы выполнялось условие ${\displaystyle (x_{1},x_{2},x_{3},x_{4})=1}$.

Коровьев, 2012 год

• ${\displaystyle x=-(2a^{2}-2ab-b^{2})cd^{3}-(a^{2}-ab+b^{2})^{2}c^{4}}$

  ${\displaystyle y=\quad (2a^{2}-2ab-b^{2})c^{3}d+(a^{2}-ab+b^{2})^{2}d^{4}}$

  ${\displaystyle z=\quad (a^{2}+2ab-2b^{2})c^{3}d-(a^{2}-ab+b^{2})^{2}d^{4}}$

  ${\displaystyle w=\quad (a^{2}+2ab-2b^{2})cd^{3}-(a^{2}-ab+b^{2})^{2}c^{4}}$

где ${\displaystyle a}$, ${\displaystyle b\,,\,c}$ и ${\displaystyle d}$ — любые целые числа^[8].