Задача про чотири куби полягає в знаходженні всіх цілочисельних розв'язків діофантового рівняння :
![{\displaystyle x^{3}+y^{3}+z^{3}=w^{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7063b53c8e408361d59814872c99e77d4da46f0)
Слід зазначити, що попри те, що запропоновано кілька повних розв'язків цього рівняння в раціональних числах, його повний розв'язок у цілих числах на 2018 рік невідомий[1].
Ще Платон знав, що сума кубів сторін піфагорійського трикутника також є кубом
[2], про що він згадує в своїй «Державі»[3].
Найменші натуральні розв'язки:
![{\displaystyle 3^{3}+4^{3}+5^{3}=6^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed36f1e006eee60cd22f517d51a1537591af1dc4)
![{\displaystyle 1^{3}+6^{3}+8^{3}=9^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24ceb6cc63623acd2e1c8f96d4b9db4ead2d0c5e)
![{\displaystyle 3^{3}+10^{3}+18^{3}=19^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/489b5614c8dab7b575c32bbfea73331b9069ab64)
![{\displaystyle 7^{3}+14^{3}+17^{3}=20^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec95e32f929647039f024f7c725d68c3ef4935)
![{\displaystyle 4^{3}+17^{3}+22^{3}=25^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65f415df1b5e0d6bdd264a1e6106747f7d03304a)
![{\displaystyle 18^{3}+19^{3}+21^{3}=28^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/658a6d83a83162224d1a02c32a452076f9f2d65b)
![{\displaystyle 11^{3}+15^{3}+27^{3}=29^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cdabf45b31f65f8e6d6004ee387ffce25b66d7)
![{\displaystyle 2^{3}+17^{3}+40^{3}=41^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50a8b70a7be7ab650fe7f5ed5c5122bef561e2ad)
![{\displaystyle 6^{3}+32^{3}+33^{3}=41^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb40d0f4968f83fd584152959b260d3321ed90b)
![{\displaystyle 16^{3}+23^{3}+41^{3}=44^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/526f8287335425985f46b153218e3aff3a788cfa)
Якщо дозволити від'ємні значення, то мають місце рівності:
![{\displaystyle -1^{3}+9^{3}+10^{3}=12^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af44088db4555f17d6f1aa8ebdaf5f83fe741e36)
![{\displaystyle -2^{3}+9^{3}+15^{3}=16^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92737a4a91483eb6194d2300038f1927046573a4)
![{\displaystyle -2^{3}+15^{3}+33^{3}=34^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c25913912c41bc53b14c990f3f858f1bb63da306)
![{\displaystyle -2^{3}+41^{3}+86^{3}=89^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5909bf5ac8c3e3a035303db6aa1f1e5b90ee3b1c)
![{\displaystyle -3^{3}+22^{3}+59^{3}=60^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af010a65ff0d2d0a1db592d33601311619ad1128)
- Ґ. Гарді і Райт (1938)[4][5]
![{\displaystyle x=-a(b-3c)(b^{2}+3c^{2})+a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7c4e48a656f945ca6fde96dcea56a64a96c2da)
![{\displaystyle y=\quad a(b+3c)(b^{2}+3c^{2})-a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97100390ff0819bc6b193760009cd12cda9e6332)
![{\displaystyle z=\quad a^{3}(b-3c)-(b^{2}+3c^{2})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13e85a1f96287959403b58a29b7719a89cfb59c0)
![{\displaystyle w=\quad a^{3}(b+3c)-(b^{2}+3c^{2})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b157d92e241d53d08e686daba41331f156cc390f)
- Н. Елкіс[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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd64834e83b596e49fb79ab91f9660175b63fbf4)
- Леонард Ейлер (1740)
![{\displaystyle x=1-(a-3b)(a^{2}+3b^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10793a9c3afca240da55c097193a40a634a68577)
![{\displaystyle y=-1+(a+3b)(a^{2}+3b^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11ad49c6197c7b50f11faa4aaed69a9a9af40886)
![{\displaystyle z=-a-3b+(a^{2}+3b^{2})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ba3a47514edcbc217c4690ee1871083dce0f14)
![{\displaystyle w=-a+3b+(a^{2}+3b^{2})^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92bae4338a0d459efdc3e2034fa55c85870aa295)
- Линник (1940)
![{\displaystyle x=b(a^{6}-b^{6})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b5615797a232d56653a4c36b58d386ecf28df4)
![{\displaystyle y=a(a^{6}-b^{6})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1526ec8e285a13eef6add62c95e1b34bc20c6b18)
![{\displaystyle z=b(2a^{6}+3a^{3}b^{3}+b^{6})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd07291ab17675c1c69dfbb18b63b62c39f52e63)
![{\displaystyle w=a(a^{6}+3a^{3}b^{3}+2b^{6})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9894145ee8974497809047dd30285f7e2967ae54)
![{\displaystyle x=a^{2}(b^{6}-7)+9ac-3c^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/128c823678a178e6af6f8b7ea5452b33604b28ed)
![{\displaystyle y=a^{2}{\big [}b^{3}(2b^{3}+9)+7{\big ]}-3ac(2b^{3}+3)+3c^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9f2a74ee05af9064a23eb6a7e492253853615c)
![{\displaystyle z=a^{2}b{\big [}b^{3}(b^{3}+3)+2{\big ]}-3abc(b^{3}+2)+3bc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a4ccec2cfd85a91f792192f25547534fedeecb)
![{\displaystyle w=a^{2}b{\big [}b^{3}(b^{3}+6)+11{\big ]}-3abc(b^{3}+4)+3bc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e16f6c1b0d1c3ddaba95512140fb6ec21a9ec198)
![{\displaystyle x=3a^{2}(b^{6}-7)-9ac-c^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6705e0a1f4a07343f18a6a8d26593f0d5685dfca)
![{\displaystyle y=3a^{2}{\big [}b^{3}(2b^{3}-9)+7{\big ]}-3ac(2b^{3}-3)+c^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c239f633c5e3fac21159d06f48a09fc549df10d)
![{\displaystyle z=3a^{2}b{\big [}b^{3}(b^{3}-6)+11{\big ]}-3abc(b^{3}-4)+bc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7309f88ad7f047def094988fe780279705ad2d)
![{\displaystyle w=3a^{2}b{\big [}b^{3}(b^{3}-3)+2{\big ]}-3abc(b^{3}-2)+bc^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f85284d8da305074bdb6b5f9a1f4a583def725)
- Roger Heath-Brown [1] [Архівовано 21 січня 2022 у Wayback Machine.] (1993)
![{\displaystyle x=9a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f4c1a562616f85cc10b68a73f682b88cef639f)
![{\displaystyle y=3a-9a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf85c422d6aece1966df10ade5190f68d3ab619)
![{\displaystyle z=1-9a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bceacba8b5a04fae95a3f3142cf3532aaf7b6f0a)
![{\displaystyle w=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19cbe03d2cb784a6fa6cd3727d4d2a71ed46fb74)
- Луїс Морделл[ru] (1956)
![{\displaystyle x=9a^{3}b+b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab1a89d9961ea6120c08be4a3694804ea4c468e)
![{\displaystyle y=9a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf98a2eee033359527539fd580666c1035f082e)
![{\displaystyle z=-b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00cf965ba1959e5429fc18c4bcd967438437eb58)
![{\displaystyle w=9a^{4}+3ab^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4de15cb0058b007417837f6f7316fcd2790692)
![{\displaystyle x=9a^{3}b-b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57008a2a03eff69dc49081cddad8ba1c2b5c3399)
![{\displaystyle y=9a^{4}-3ab^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/812853c45d6a9578c535393e821c24bb302afb56)
![{\displaystyle z=b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d99943e2b0ec6fcd54b97853c62ec9527be21e0)
![{\displaystyle w=9a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90c7b38c9af7ced55bfa16b9425950dbce55acef)
![{\displaystyle x=9a^{3}b+b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab1a89d9961ea6120c08be4a3694804ea4c468e)
![{\displaystyle y=9a^{3}b-b^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a943be9550ed058d233bf9b9d792165ea6a4750)
![{\displaystyle z=9a^{4}-3ab^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/428e0991b81d1be7daa72ce23ce3b8def449ac9a)
![{\displaystyle w=9a^{4}+3ab^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4de15cb0058b007417837f6f7316fcd2790692)
- Розв'язок, отриманий методом алгебричної геометрії
![{\displaystyle x=3a\left(a^{2}+ab+b^{2}\right)-9}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f6b10001f3a26f8f5b468a72cf83da64369474)
![{\displaystyle y=\left(a^{2}+ab+b^{2}\right)^{2}-9a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13d1c437d8a51a57ee0fda0821fbb0e0b4f3b00)
![{\displaystyle z=3\left(a^{2}+ab+b^{2}\right)(a+b)+9}](https://wikimedia.org/api/rest_v1/media/math/render/svg/215cd9891b9db0e24acaf6864042e8e93750ff63)
![{\displaystyle w=\left(a^{2}+ab+b^{2}\right)^{2}+9(a+b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a567cdab31ed8b8f26bbf8424ec3b405565f5eb9)
- Рамануджан
![{\displaystyle x=3a^{2}+5ab-5b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c29c35d12ed1301ce43b1e668f20efbaaef9a8)
![{\displaystyle y=4a^{2}-4ab+6b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/763de2e0d52d41be188eac28f86b8543beb3b2dc)
![{\displaystyle z=5a^{2}-5ab-3b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01bde6b349bc63b5733c9a660c30f220d853cb40)
![{\displaystyle w=6a^{2}-4ab+4b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f99e400fec24bae665eba80e8d98c666c947b71)
![{\displaystyle x=a^{7}-3a^{4}(1+b)+a(2+6b+3b^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dde2d1cd60ee916d48da817dfacc2810aea14a3)
![{\displaystyle y=2a^{6}-3a^{3}(1+2b)+1+3b+3b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3cba704f7b51c747be9d29d248ee04ee685b733)
![{\displaystyle z=a^{6}-1-3b-3b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df278eed01da3add9b3650b93a83c7a440097b69)
![{\displaystyle w=a^{7}-3a^{4}b+a(3b^{2}-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f28b13603a1b044581e52031f19b2846bbe51e31)
![{\displaystyle x=-a^{2}+9ab+b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22361f0e9c3022aaaa586a51e278a39484256c6a)
![{\displaystyle y=a^{2}+7ab-9b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb0aad53b4e6cd396a975b833681ba7e000001c)
![{\displaystyle z=2a^{2}-4ab+12b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ec0c24420854ea26480031221e953651ab1d8e)
![{\displaystyle w=2a^{2}+10b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a30d335d5dd26682078a2303dbc995b32ab347)
- Невідомий автор (1825)
![{\displaystyle x=a^{9}-3^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/678f96a04ccfe4a69afc79c98b9656853aadff5a)
![{\displaystyle y=-a^{9}+3^{5}a^{3}+3^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26227de96fbe2b37bd582b8a7e28e4cd4321cc65)
![{\displaystyle z=3^{3}a^{6}+3^{5}a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7516ff73689ab66381d96ed7665de1b459c4a6ea)
![{\displaystyle w=3^{2}a^{7}+3^{4}a^{4}+3^{6}a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06824d62a4f4c040e5fb26eb6b7d412e3562872e)
- Деррик Лемер[ru] (1955)
![{\displaystyle x=3888a^{10}-135a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ce797736db1c93da6e43805edd6f0be6cf9705)
![{\displaystyle y=-3888a^{10}-1296a^{7}-81a^{4}+3a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc097a397b00dd526c625790f7b3c88bd7a7b87c)
![{\displaystyle z=3888a^{9}+648a^{6}-9a^{3}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10112f47931a9ce6da79d959e5c67ec4e548ceb5)
![{\displaystyle w=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19cbe03d2cb784a6fa6cd3727d4d2a71ed46fb74)
- В. Б. Лабковський
![{\displaystyle x=4b^{2}-11b-21}](https://wikimedia.org/api/rest_v1/media/math/render/svg/686cd36aef1b457f19d851996b394b024c44e8be)
![{\displaystyle y=3b^{2}+11b-28}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9b1f1621d85fa7c72dec3504c24653dcc832a2)
![{\displaystyle z=5b^{2}-7b+42}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10a2ab45713fe7de6a1522a9d97dbfd874812354)
![{\displaystyle w=6b^{2}-7b+35}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9064eb3245dc3854f814550a463ee0e0aa1e602a)
- Гарді і Райт
![{\displaystyle x=a(a^{3}-2b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46104b9f41b111a2aaa0cd50572d66469e49ec8a)
![{\displaystyle y=b(2a^{3}-b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/678d6c3e8b4b1cad43c6f5b89b6651dbfca86eb5)
![{\displaystyle z=b(a^{3}+b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7efc3fb68997b507d74530710e2806a1dfb2923)
![{\displaystyle w=a(a^{3}+b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53fc2933d3b60d0fa349d14191db815f6a556b9f)
![{\displaystyle x=a(a^{3}-b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2584ff5771c16cf0a2abf0eca4eebcbaf89db6e)
![{\displaystyle y=b(a^{3}-b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4901a2e8591b2058bb58755772792487afcf96d)
![{\displaystyle z=b(2a^{3}+b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f38000f1ac62b84edf69e75b7406aabde9cdb13)
![{\displaystyle w=a(a^{3}+2b^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17734c0f25efc696998ee13081ac85c99dafc85)
- Г. Александров (1972)
![{\displaystyle x=7a^{2}+17ab-6b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/486b6a7b8b357bca652117edbd2165276f026fe2)
![{\displaystyle y=42a^{2}-17ab-b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c052d1365a33e288dc18c5202a2eee3d577affe)
![{\displaystyle z=56a^{2}-35ab+9b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a5d0c632eb96166f44e065226a69916034f8ec)
![{\displaystyle w=63a^{2}-35ab+8b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db57011a029274d5a983bc4d229a226f136dc3c9)
![{\displaystyle x=7a^{2}+17ab-17b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b05494dddb0bac206940514b040fa2cd0122c9)
![{\displaystyle y=17a^{2}-17ab-7b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b608f7659984ac932979e0d5296ae9d7e777f18)
![{\displaystyle z=14a^{2}-20ab+20b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49674350cfa59a76e706fd1a1df58752c67fd623)
![{\displaystyle w=20a^{2}-20ab+14b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff30c7a2e73a210f7c6351888807ebc769b0fcb9)
![{\displaystyle x=21a^{2}+23ab-19b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f7437723d16311970b32a34de0e5a1d5798fd0)
![{\displaystyle y=19a^{2}-23ab-21b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d9e2437018e678b86e3de547384d15763d68f9)
![{\displaystyle z=18a^{2}+4ab+28b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbfec440978c51a42018dcf6b88c2d91633fe37)
![{\displaystyle w=28a^{2}+4ab+18b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd989641d9ced6bbdda4808040019ef554f2e32)
![{\displaystyle x=3a^{2}+41ab-37b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c29b8c0a9abe8c0cd4972db9d7966ec8ce35ff8)
![{\displaystyle y=37a^{2}-41ab-3b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d55766988a5d2d8d0686c3ee924ca972e956115e)
![{\displaystyle z=36a^{2}-68ab+46b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13b99ae31e8bd9d57c55fa58a3465aede75b565c)
![{\displaystyle w=46a^{2}-68ab+36b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60bf03ce8d585ff6a3cfb211dfed8511ed06ef8d)
![{\displaystyle x=-4a^{2}+22ab-9b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7cf588f936cc5985a7fdba71ebe8aefe6217b1)
![{\displaystyle y=36a^{2}-22ab+b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0f763cc7238e11ed0a082b597b282cafd22359)
![{\displaystyle z=40a^{2}-40ab+12b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e808d76daf8d28763feaafad23e1275538da74)
![{\displaystyle w=48a^{2}-40ab+10b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49335897e2be44f63f203e3705d784b5404b9669)
- Ajai Choudhry (1998)[6]
![{\displaystyle dx_{1}=(a^{4}+2a^{3}b+3a^{2}b^{2}+2ab^{3}+b^{4})+(2a+b)c^{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1deda7a139982851f203ae70438acb9e94b78b4b)
![{\displaystyle dx_{2}=-\{a^{4}+2a^{3}b+3a^{2}b^{2}+2ab^{3}+b^{4}-(a-b)c^{3}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82124cafdedaad2ce13d02d7e745a02ac748ab43)
![{\displaystyle dx_{3}=c(-a^{3}+b^{3}+c^{3}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b69f1df4bd9474559e3d1d5bb3a2f7c12629404)
![{\displaystyle dx_{4}=-\{(2a^{3}+3a^{2}b+3ab^{2}+b^{3})c+c^{4}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04ccd79516f40d13dad6c08fbaaaba0a0cc5c93)
де числа
— довільні цілі, а число
вибрано так, щоб виконувалася умова
.
- Коров'єв (2012)
![{\displaystyle x=-(2a^{2}-2ab-b^{2})cd^{3}-(a^{2}-ab+b^{2})^{2}c^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b38995169d2e960e334d5941a5eb497f6db4c19f)
![{\displaystyle y=\quad (2a^{2}-2ab-b^{2})c^{3}d+(a^{2}-ab+b^{2})^{2}d^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6f6ae06fe46a5aeaab1c154e30ffdff32343fe)
![{\displaystyle z=\quad (a^{2}+2ab-2b^{2})c^{3}d-(a^{2}-ab+b^{2})^{2}d^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/767381f2be38f8206686b99f507208a1b383dea6)
![{\displaystyle w=\quad (a^{2}+2ab-2b^{2})cd^{3}-(a^{2}-ab+b^{2})^{2}c^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc6eabe9be7709a8dd02e3618a222d90f5cad1f)
де
,
і
— будь-які цілі числа.[7]
- ↑ а б Cohen, Henri[en]. 6.4 Diophantine Equations of Degree 3 // Number Theory – Volume I: Tools and Diophantine Equations. — Springer-Verlag, 2007. — Т. 239. — (Graduate Texts in Mathematics) — ISBN 978-0-387-49922-2.
- ↑ Перельман Я.И. Занимательная алгебра / Под редакцией и с дополнениями В.Г. Болтянского. — Издание одиннадцатое. — Москва : Издательство «Наука»: Главная редакция физико-математической литературы, 1967. — С. 120—121.
- ↑ Марио Ливио. φ – Число Бога. Золотое сечение – формула мироздания. — АСТ, 2015. — С. 110. — ISBN 978-5-17-094497-2.
- ↑ An introduction to the theory of numbers. — First ed. — Oxford : Oxford University Press, 1938.
- ↑ Цитата из раздела «1.3.7 Уравнение
» из книги Харди и Райта - ↑ Ajai Choudhry. On Equal Sums of Cubes [Архівовано 21 липня 2020 у Wayback Machine.]. Rocky Mountain J. Math. Volume 28, Number 4 (1998), 1251—1257.
- ↑ У багатьох випадках числа
мають спільні дільники. Щоб отримати примітивну четвірку чисел, досить скоротити кожне з чисел на їхній найбільший спільний дільник.