Zadacha pro chotiri kubi polyagaye v znahodzhenni vsih cilochiselnih rozv yazkiv diofantovogo rivnyannya x 3 y 3 z 3 w 3 displaystyle x 3 y 3 z 3 w 3 Slid zaznachiti sho popri te sho zaproponovano kilka povnih rozv yazkiv cogo rivnyannya v racionalnih chislah jogo povnij rozv yazok u cilih chislah na 2018 rik nevidomij 1 Zmist 1 Istoriya 2 Prikladi cilochiselnih rozv yazkiv 3 Povni racionalni parametrizaciyi 4 Inshi seriyi rozv yazkiv 5 Div takozh 6 Primitki 7 LiteraturaIstoriya RedaguvatiShe Platon znav sho suma kubiv storin pifagorijskogo trikutnika takozh ye kubom 3 3 4 3 5 3 6 3 displaystyle 3 3 4 3 5 3 6 3 nbsp 2 pro sho vin zgaduye v svoyij Derzhavi 3 Prikladi cilochiselnih rozv yazkiv RedaguvatiNajmenshi naturalni rozv yazki 3 3 4 3 5 3 6 3 displaystyle 3 3 4 3 5 3 6 3 nbsp 1 3 6 3 8 3 9 3 displaystyle 1 3 6 3 8 3 9 3 nbsp 3 3 10 3 18 3 19 3 displaystyle 3 3 10 3 18 3 19 3 nbsp 7 3 14 3 17 3 20 3 displaystyle 7 3 14 3 17 3 20 3 nbsp 4 3 17 3 22 3 25 3 displaystyle 4 3 17 3 22 3 25 3 nbsp 18 3 19 3 21 3 28 3 displaystyle 18 3 19 3 21 3 28 3 nbsp 11 3 15 3 27 3 29 3 displaystyle 11 3 15 3 27 3 29 3 nbsp 2 3 17 3 40 3 41 3 displaystyle 2 3 17 3 40 3 41 3 nbsp 6 3 32 3 33 3 41 3 displaystyle 6 3 32 3 33 3 41 3 nbsp 16 3 23 3 41 3 44 3 displaystyle 16 3 23 3 41 3 44 3 nbsp dd Yaksho dozvoliti vid yemni znachennya to mayut misce rivnosti 1 3 9 3 10 3 12 3 displaystyle 1 3 9 3 10 3 12 3 nbsp 2 3 9 3 15 3 16 3 displaystyle 2 3 9 3 15 3 16 3 nbsp 2 3 15 3 33 3 34 3 displaystyle 2 3 15 3 33 3 34 3 nbsp 2 3 41 3 86 3 89 3 displaystyle 2 3 41 3 86 3 89 3 nbsp 3 3 22 3 59 3 60 3 displaystyle 3 3 22 3 59 3 60 3 nbsp dd Povni racionalni parametrizaciyi RedaguvatiG Gardi i Rajt 1938 4 5 x a b 3 c b 2 3 c 2 a 4 displaystyle x a b 3c b 2 3c 2 a 4 nbsp y a b 3 c b 2 3 c 2 a 4 displaystyle y quad a b 3c b 2 3c 2 a 4 nbsp z a 3 b 3 c b 2 3 c 2 2 displaystyle z quad a 3 b 3c b 2 3c 2 2 nbsp w a 3 b 3 c b 2 3 c 2 2 displaystyle w quad a 3 b 3c b 2 3c 2 2 nbsp N Elkis 1 x d s r t 2 s 2 2 r 2 t s 3 r s 2 2 r 2 s r 3 y d t 3 s r t 2 s 2 2 r 2 t r s 2 2 r 2 s r 3 z d t 3 s r t 2 s 2 2 r 2 t 2 r s 2 r 2 s 2 r 3 w d s 2 r t 2 r 2 s 2 t s 3 r s 2 2 r 2 s 2 r 3 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 nbsp Inshi seriyi rozv yazkiv RedaguvatiLeonard Ejler 1740 x 1 a 3 b a 2 3 b 2 displaystyle x 1 a 3b a 2 3b 2 nbsp y 1 a 3 b a 2 3 b 2 displaystyle y 1 a 3b a 2 3b 2 nbsp z a 3 b a 2 3 b 2 2 displaystyle z a 3b a 2 3b 2 2 nbsp w a 3 b a 2 3 b 2 2 displaystyle w a 3b a 2 3b 2 2 nbsp Linnik 1940 x b a 6 b 6 displaystyle x b a 6 b 6 nbsp y a a 6 b 6 displaystyle y a a 6 b 6 nbsp z b 2 a 6 3 a 3 b 3 b 6 displaystyle z b 2a 6 3a 3 b 3 b 6 nbsp w a a 6 3 a 3 b 3 2 b 6 displaystyle w a a 6 3a 3 b 3 2b 6 nbsp x a 2 b 6 7 9 a c 3 c 2 displaystyle x a 2 b 6 7 9ac 3c 2 nbsp y a 2 b 3 2 b 3 9 7 3 a c 2 b 3 3 3 c 2 displaystyle y a 2 big b 3 2b 3 9 7 big 3ac 2b 3 3 3c 2 nbsp z a 2 b b 3 b 3 3 2 3 a b c b 3 2 3 b c 2 displaystyle z a 2 b big b 3 b 3 3 2 big 3abc b 3 2 3bc 2 nbsp w a 2 b b 3 b 3 6 11 3 a b c b 3 4 3 b c 2 displaystyle w a 2 b big b 3 b 3 6 11 big 3abc b 3 4 3bc 2 nbsp x 3 a 2 b 6 7 9 a c c 2 displaystyle x 3a 2 b 6 7 9ac c 2 nbsp y 3 a 2 b 3 2 b 3 9 7 3 a c 2 b 3 3 c 2 displaystyle y 3a 2 big b 3 2b 3 9 7 big 3ac 2b 3 3 c 2 nbsp z 3 a 2 b b 3 b 3 6 11 3 a b c b 3 4 b c 2 displaystyle z 3a 2 b big b 3 b 3 6 11 big 3abc b 3 4 bc 2 nbsp w 3 a 2 b b 3 b 3 3 2 3 a b c b 3 2 b c 2 displaystyle w 3a 2 b big b 3 b 3 3 2 big 3abc b 3 2 bc 2 nbsp Roger Heath Brown 1 Arhivovano 21 sichnya 2022 u Wayback Machine 1993 x 9 a 4 displaystyle x 9a 4 nbsp y 3 a 9 a 4 displaystyle y 3a 9a 4 nbsp z 1 9 a 3 displaystyle z 1 9a 3 nbsp w 1 displaystyle w 1 nbsp Luyis Mordell ru 1956 x 9 a 3 b b 4 displaystyle x 9a 3 b b 4 nbsp y 9 a 4 displaystyle y 9a 4 nbsp z b 4 displaystyle z b 4 nbsp w 9 a 4 3 a b 3 displaystyle w 9a 4 3ab 3 nbsp x 9 a 3 b b 4 displaystyle x 9a 3 b b 4 nbsp y 9 a 4 3 a b 3 displaystyle y 9a 4 3ab 3 nbsp z b 4 displaystyle z b 4 nbsp w 9 a 4 displaystyle w 9a 4 nbsp x 9 a 3 b b 4 displaystyle x 9a 3 b b 4 nbsp y 9 a 3 b b 4 displaystyle y 9a 3 b b 4 nbsp z 9 a 4 3 a b 3 displaystyle z 9a 4 3ab 3 nbsp w 9 a 4 3 a b 3 displaystyle w 9a 4 3ab 3 nbsp Rozv yazok otrimanij metodom algebrichnoyi geometriyix 3 a a 2 a b b 2 9 displaystyle x 3a left a 2 ab b 2 right 9 nbsp y a 2 a b b 2 2 9 a displaystyle y left a 2 ab b 2 right 2 9a nbsp z 3 a 2 a b b 2 a b 9 displaystyle z 3 left a 2 ab b 2 right a b 9 nbsp w a 2 a b b 2 2 9 a b displaystyle w left a 2 ab b 2 right 2 9 a b nbsp Ramanudzhanx 3 a 2 5 a b 5 b 2 displaystyle x 3a 2 5ab 5b 2 nbsp y 4 a 2 4 a b 6 b 2 displaystyle y 4a 2 4ab 6b 2 nbsp z 5 a 2 5 a b 3 b 2 displaystyle z 5a 2 5ab 3b 2 nbsp w 6 a 2 4 a b 4 b 2 displaystyle w 6a 2 4ab 4b 2 nbsp x a 7 3 a 4 1 b a 2 6 b 3 b 2 displaystyle x a 7 3a 4 1 b a 2 6b 3b 2 nbsp y 2 a 6 3 a 3 1 2 b 1 3 b 3 b 2 displaystyle y 2a 6 3a 3 1 2b 1 3b 3b 2 nbsp z a 6 1 3 b 3 b 2 displaystyle z a 6 1 3b 3b 2 nbsp w a 7 3 a 4 b a 3 b 2 1 displaystyle w a 7 3a 4 b a 3b 2 1 nbsp x a 2 9 a b b 2 displaystyle x a 2 9ab b 2 nbsp y a 2 7 a b 9 b 2 displaystyle y a 2 7ab 9b 2 nbsp z 2 a 2 4 a b 12 b 2 displaystyle z 2a 2 4ab 12b 2 nbsp w 2 a 2 10 b 2 displaystyle w 2a 2 10b 2 nbsp Nevidomij avtor 1825 x a 9 3 6 displaystyle x a 9 3 6 nbsp y a 9 3 5 a 3 3 6 displaystyle y a 9 3 5 a 3 3 6 nbsp z 3 3 a 6 3 5 a 3 displaystyle z 3 3 a 6 3 5 a 3 nbsp w 3 2 a 7 3 4 a 4 3 6 a displaystyle w 3 2 a 7 3 4 a 4 3 6 a nbsp Derrik Lemer ru 1955 x 3888 a 10 135 a 4 displaystyle x 3888a 10 135a 4 nbsp y 3888 a 10 1296 a 7 81 a 4 3 a displaystyle y 3888a 10 1296a 7 81a 4 3a nbsp z 3888 a 9 648 a 6 9 a 3 1 displaystyle z 3888a 9 648a 6 9a 3 1 nbsp w 1 displaystyle w 1 nbsp V B Labkovskijx 4 b 2 11 b 21 displaystyle x 4b 2 11b 21 nbsp y 3 b 2 11 b 28 displaystyle y 3b 2 11b 28 nbsp z 5 b 2 7 b 42 displaystyle z 5b 2 7b 42 nbsp w 6 b 2 7 b 35 displaystyle w 6b 2 7b 35 nbsp Gardi i Rajtx a a 3 2 b 3 displaystyle x a a 3 2b 3 nbsp y b 2 a 3 b 3 displaystyle y b 2a 3 b 3 nbsp z b a 3 b 3 displaystyle z b a 3 b 3 nbsp w a a 3 b 3 displaystyle w a a 3 b 3 nbsp x a a 3 b 3 displaystyle x a a 3 b 3 nbsp y b a 3 b 3 displaystyle y b a 3 b 3 nbsp z b 2 a 3 b 3 displaystyle z b 2a 3 b 3 nbsp w a a 3 2 b 3 displaystyle w a a 3 2b 3 nbsp G Aleksandrov 1972 x 7 a 2 17 a b 6 b 2 displaystyle x 7a 2 17ab 6b 2 nbsp y 42 a 2 17 a b b 2 displaystyle y 42a 2 17ab b 2 nbsp z 56 a 2 35 a b 9 b 2 displaystyle z 56a 2 35ab 9b 2 nbsp w 63 a 2 35 a b 8 b 2 displaystyle w 63a 2 35ab 8b 2 nbsp x 7 a 2 17 a b 17 b 2 displaystyle x 7a 2 17ab 17b 2 nbsp y 17 a 2 17 a b 7 b 2 displaystyle y 17a 2 17ab 7b 2 nbsp z 14 a 2 20 a b 20 b 2 displaystyle z 14a 2 20ab 20b 2 nbsp w 20 a 2 20 a b 14 b 2 displaystyle w 20a 2 20ab 14b 2 nbsp x 21 a 2 23 a b 19 b 2 displaystyle x 21a 2 23ab 19b 2 nbsp y 19 a 2 23 a b 21 b 2 displaystyle y 19a 2 23ab 21b 2 nbsp z 18 a 2 4 a b 28 b 2 displaystyle z 18a 2 4ab 28b 2 nbsp w 28 a 2 4 a b 18 b 2 displaystyle w 28a 2 4ab 18b 2 nbsp x 3 a 2 41 a b 37 b 2 displaystyle x 3a 2 41ab 37b 2 nbsp y 37 a 2 41 a b 3 b 2 displaystyle y 37a 2 41ab 3b 2 nbsp z 36 a 2 68 a b 46 b 2 displaystyle z 36a 2 68ab 46b 2 nbsp w 46 a 2 68 a b 36 b 2 displaystyle w 46a 2 68ab 36b 2 nbsp x 4 a 2 22 a b 9 b 2 displaystyle x 4a 2 22ab 9b 2 nbsp y 36 a 2 22 a b b 2 displaystyle y 36a 2 22ab b 2 nbsp z 40 a 2 40 a b 12 b 2 displaystyle z 40a 2 40ab 12b 2 nbsp w 48 a 2 40 a b 10 b 2 displaystyle w 48a 2 40ab 10b 2 nbsp Ajai Choudhry 1998 6 d x 1 a 4 2 a 3 b 3 a 2 b 2 2 a b 3 b 4 2 a b c 3 displaystyle dx 1 a 4 2a 3 b 3a 2 b 2 2ab 3 b 4 2a b c 3 nbsp d x 2 a 4 2 a 3 b 3 a 2 b 2 2 a b 3 b 4 a b c 3 displaystyle dx 2 a 4 2a 3 b 3a 2 b 2 2ab 3 b 4 a b c 3 nbsp d x 3 c a 3 b 3 c 3 displaystyle dx 3 c a 3 b 3 c 3 nbsp d x 4 2 a 3 3 a 2 b 3 a b 2 b 3 c c 4 displaystyle dx 4 2a 3 3a 2 b 3ab 2 b 3 c c 4 nbsp de chisla a b c displaystyle a b c nbsp dovilni cili a chislo d 0 displaystyle d neq 0 nbsp vibrano tak shob vikonuvalasya umova x 1 x 2 x 3 x 4 1 displaystyle x 1 x 2 x 3 x 4 1 nbsp Korov yev 2012 x 2 a 2 2 a b b 2 c d 3 a 2 a b b 2 2 c 4 displaystyle x 2a 2 2ab b 2 cd 3 a 2 ab b 2 2 c 4 nbsp y 2 a 2 2 a b b 2 c 3 d a 2 a b b 2 2 d 4 displaystyle y quad 2a 2 2ab b 2 c 3 d a 2 ab b 2 2 d 4 nbsp z a 2 2 a b 2 b 2 c 3 d a 2 a b b 2 2 d 4 displaystyle z quad a 2 2ab 2b 2 c 3 d a 2 ab b 2 2 d 4 nbsp w a 2 2 a b 2 b 2 c d 3 a 2 a b b 2 2 c 4 displaystyle w quad a 2 2ab 2b 2 cd 3 a 2 ab b 2 2 c 4 nbsp de a displaystyle a nbsp b c displaystyle b c nbsp i d displaystyle d nbsp bud yaki cili chisla 7 Div takozh RedaguvatiGipoteza Ejlera Suma troh kubivPrimitki Redaguvati a b Cohen Henri en 6 4 Diophantine Equations of Degree 3 Number Theory Volume I Tools and Diophantine Equations Springer Verlag 2007 T 239 Graduate Texts in Mathematics ISBN 978 0 387 49922 2 Perelman Ya I Zanimatelnaya algebra Pod redakciej i s dopolneniyami V G Boltyanskogo Izdanie odinnadcatoe Moskva Izdatelstvo Nauka Glavnaya redakciya fiziko matematicheskoj literatury 1967 S 120 121 Mario Livio f Chislo Boga Zolotoe sechenie formula mirozdaniya AST 2015 S 110 ISBN 978 5 17 094497 2 An introduction to the theory of numbers First ed Oxford Oxford University Press 1938 Citata iz razdela 1 3 7 Uravnenie x 3 y 3 z 3 t 3 displaystyle x 3 y 3 z 3 t 3 nbsp iz knigi Hardi i Rajta Ajai Choudhry On Equal Sums of Cubes Arhivovano 21 lipnya 2020 u Wayback Machine Rocky Mountain J Math Volume 28 Number 4 1998 1251 1257 U bagatoh vipadkah chisla x y z w displaystyle x y z w nbsp mayut spilni dilniki Shob otrimati primitivnu chetvirku chisel dosit skorotiti kozhne z chisel na yihnij najbilshij spilnij dilnik Literatura RedaguvatiHardi G Dvenadcat lekcij o Ramanudzhane M Institut kompyuternyh issledovanij 2002 336 s V Serpinskij 15 Reshenie uravnenij v racionalnyh chislah 2 M Fizmatlit 1961 88 s Arhivovano z dzherela 18 lipnya 2020 E Rowland Known families of integer solutions to x 3 y 3 z 3 n displaystyle x 3 y 3 z 3 n nbsp Arhivovano z dzherela 27 veresnya 2013 Reshenie Labkovskogo Zadanie 2 Arhivovano 28 veresnya 2019 u Wayback Machine Sizyj S V 20 Sravneniya lyuboj stepeni po prostomu modulyu 3 Ekaterinburg Uralskij gosudarstvennyj universitet im A M Gorkogo 1999 Arhivovano z dzherela 1 lyutogo 2019 Weisstein Eric W Diophantine Equation 3rd Powers angl na sajte Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Zadacha pro chotiri kubi amp oldid 36241486
