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Congetture  Teoria dei numeri 

Una generalizzazione dell’equazione di Fermat è l’equazione xn + ym = zp, della quale si cercano le soluzioni intere positive, fissati gli esponenti.

 

Se x, y e z possono avere un divisore comune, è facile costruire infinite soluzioni: scegliamo tre interi a, b e c tali che a + b = c e prendiamo q multiplo di mp e tale che diviso per n dia resto n – 1; r multiplo di np e tale che diviso per m dia resto m – 1 e s multiplo di nm e tale che diviso per p dia resto p – 1; una soluzione sarà allora data da Formula per x, Formula per y, e Formula per z.

Infatti xn = aq + 1brbs, ym = aqbr + 1cs, zp = aqbrcs + 1 e xn + ym = aqbrcs(a + b) = aqbrcs + 1 = zp.

Per esempio, prendiamo arbitrariamente n = 3, m = 5, p = 7 e scegliamo a = 1, b = 2, c = 3, q = 35, r = 84 e s = 90: allora x = 228330 = 55268479930183339474944, y = 217318 = 50779978334208, z = 212313 = 6530347008 e 552684799301833394749443 + 507799783342085 = 65303470087.

 

Se x, y e z non hanno un divisore comune, bisogna distinguere tre casi.

Se Formula per la definizione del caso detto “iperbolico” (caso detto “iperbolico”), la congettura di Fermat – Catalan è che esista un numero finito di soluzioni; in tutte le 10 soluzioni note un esponente è 2 (v. congettura di Fermat – Catalan).

Se Formula per la definizione del caso detto “parabolico” (caso detto “parabolico” o “euclideo”), esiste una sola soluzione: 16 + 23 = 32.

Se Formula per la definizione del caso detto “ellittico” (caso detto “ellittico” o “sferico”), escluso il caso banale nel quale uno degli esponenti è 1, F. Beukers dimostrò nel 1998 che esiste un numero finito di soluzioni parametriche, nelle quali x, y e z sono espresse come polinomi omogenei rispetto a due parametri. Le soluzioni possono essere raggruppate in quattro gruppi:

  • due esponenti uguali a 2 e uno maggiore o uguale a 2;

  • due esponenti uguali a 3 e uno uguale a 2;

  • esponenti uguali a 2, 3 e 4;

  • esponenti uguali a 2, 3 e 5.

 

Nel primo caso le soluzioni consistono nell’esprimere una potenza come somma o differenza di due quadrati (v. quadrati). In particolare nel caso dell’equazione x2 + y2 = zp si possono ottenere infinite soluzioniprendendo z uguale alla somma di due quadrati e sviluppando zp tramite l’identità dei due quadrati (v. quadrati). Per esempio, per p = 3 si ottiene la soluzione parametrica x = a(a2 + b2), y = b(3a2 + b2), z = a2 + b2.

 

 

 

Gli altri casi possono essere ricondotti, con opportuni cambiamenti di segno delle incognite, a quattro tipi di equazioni.

 

Nel caso dell’equazione x2 + y3 = z3 vi sono 6 soluzioni parametriche, per a e b interi primi tra loro (L.E. Mordell, 1969):

  • x = a6 – 20a3b3 – 8b6, y = –a4 – 8ab3, z = –4a3b + 4b4;

  • x = a6 + 20a3b3 – 8b6, y = –4a3b – 4b4, z = a4 – 8ab3;

  • x = 3a6 + 6a5b + 15a4b2 – 15a2b4 – 6ab5 – 3b6, y = –a4 + 4a3b + 6a2b2 + 4ab3b4, z =2a4 + 4a3b + 4ab3 + 2b4;

  • x = 3a6 + 6a5b + 15a4b2 – 15a2b4 – 6ab5 – 3b6, y = –2a4 – 4a3b – 8ab3 – 2b4, z = a4 – 4a3b – 6a2b2 – 4ab3 + b4;

  • x = 6a5b + 18ab5, y = –a4 – 6a2b2 – 3b4, z = a4 + 6a2b2 – 3b4;

  • x = 6a5b + 18ab5, y = –a4 – 6a2b2 + 3b4, z = –a4 + 6a2b2 + 3b4.

 

Nel caso dell’equazione x2 + y3 = z4 vi sono 7 soluzioni parametriche, per a e b interi primi tra loro (Zagier):

  • x = a12 + 396a8b4 – 4752a4b8 – 1728b12, y = –a8 + 168a4b4 – 144b8, z = 6a5b + 72ab5;

  • x = 27a12 + 1188a8b4 – 1584a4b8 – 64b12, y = –9a8 + 168a4b4 – 16b8, z = 18a5b + 24ab5;

  • x = a12 + 176a9b3 + 5632a3b9 – 1024b12, y = –8a7b – 112a4b4 + 256ab7, z = a6 – 40a3b3 – 32b6;

  • x = 12a11b – 44a9b3 + 792a7b5 + 2376a5b7 – 1188a3b9 + 2916ab11, y = a8 – 28a6b2 – 42a4b4 – 252a2b6 + 81b8, z = a6 + 15a4b2 – 45a2b4 – 27b6;

  • x = 3a12 + 12a11b + 66a10b2 – 44a9b3 – 99a8b4 + 792a7b5 + 924a6b6 + 2376a5b7 + 1485a4b8 – 1188a3b9 – 2046a2b10 – 156ab11 – 397b12, y = –2a8 – 8a7b – 56a5b3 + 28a4b4 + 168a3b5 + 112a2b6 + 88ab7 – 42b8, z = a6 – 6a5b – 15a4b2 – 20a3b3 + 15a2b4 – 30ab5 – 17b6;

  • x = 29a12 + 156a11b + 726a10b2 + 2420a9b3 + 4059a8b4 + 3960a7b5 + 2772a6b6 + 2376a5b7 + 3267a4b8 + 3564a3b9 + 1782a2b10 + 324ab11 – 27b12, y = –6a8 – 56a7b – 112a6b2 – 168a5b3 – 252a4b4 – 168a3b5 + 72ab7 + 18b8, z = 5a6 + 6a5b – 15a4b2 – 60a3b3 – 45a2b4 – 18ab5 – 9b6;

  • x = 76a12 + 516a11b + 2244a10b2 + 6072a9b3 + 9504a8b4 + 9504a7b5 + 7392a6b6 + 6336a5b7 + 6336a4b8 + 4928a3b9 + 2112a2b10 + 384ab11, y = –15a8 – 104a7b – 224a6b2 – 336a5b3 – 392a4b4 – 224a3b5 + 64ab7 + 16b8, z = 7a6 + 6a5b – 30a4b2 – 80a3b3 – 60a2b4 – 24ab5 – 8b6.

 

Nel caso dell’equazione x2 + y3 = –z4 vi sono 4 soluzioni parametriche, per a e b interi primi tra loro (Zagier):

  • x = a12 – 396a8b4 – 4752a4b8 + 1728b12, y = –a8 – 168a4b4 – 144b8, z = 6a5b – 72b5a;

  • x = 25a12 – 1188a8b4 – 1584a4b8 + 64b12, y = –9a8 – 168a4b4 – 16b8, z = 18a5b – 24b5a;

  • x = 12a11b + 44a9b3 + 792a7b5 – 2376a5b7 – 1188a3b9 – 2916ab11, y = –a8 – 28a6b2 + 42a4b4 – 252a2b6 – 81b8, z = a6 – 15a4b2 – 45a2b4 + 27b6;

  • x = 46a12 + 60a11b – 132a10b2 – 1012a9b3 – 2574a8b4 – 2376a7b5 – 1848a6b6 – 2376a5b7 – 2574a4b8 – 1012a3b9 – 132a2b10 + 60ab11 + 46b12, y = –13a8 – 16a7b – 28a6b2 – 112a5b3 – 238a4b4 – 112a3b5 – 28a2b6 – 16ab7 – 13b8, z = 3a6 + 24a5b + 15a4b2 – 15a2b4 – 24ab5 – 3b6.

 

Nel caso dell’equazione x2 + y3 = z5 vi sono 27 soluzioni parametriche, per a e b interi primi tra loro (F. Beukers ne trovò 22 nel 1998, poi J. Edwards completò l’elenco nel 2004):

  • x = a30 – 75168a25b5 – 207463680a20b10 –4301966868480a10b20 + 32320864124928a5b25 + 8916100448256b30, y = –a20 – 32832a15b5 – 10243584a10b10 + 680804352a5b15 – 429981696b20, z = –12a11b + 19008a6b6 + 248832ab11;

  • x = a30 – 2900a27b3 + 156600a24b6 – 32016000a21b9 – 2433216000a18b12 – 778629120000a12b18 + 3278438400000a9b21 + 5131468800000a6b24 + 30408704000000a3b27 + 3355443200000b30, y = 20a19b – 5700a16b4 – 182400a13b7 – 15808000a10b10 + 58368000a7b13 – 583680000a4b16 – 655360000ab19, z = a12 + 440a9b3 – 10560a6b6 – 140800a3b9 + 102400b12;

  • x = 30a29b – 1160a27b3 + 95526a25b5 + 1879200a23b7 + 24312150a21b9 – 162081000a19b11 + 6928962750a17b13 – 311803323750a13b17 + 328214025000a11b19 – 2215444668750a9b21 – 7705894500000a7b23 – 17627233668750a5b25 + 9632368125000a3b27 – 11210083593750ab29, y = a20 – 190a18b2 – 855a16b4 – 102600a14b6 – 1000350a12b8 + 10003500a10b10 – 45015750a8b12 – 207765000a6b14 – 77911875a4b16 – 779118750a2b18 + 184528125b20, z = a12 + 66a10b2 – 1485a8b4 – 5940a6b6 – 66825a4b8 + 133650a2b10 + 91125b12;

  • x = 30a29b + 1160a27b3 + 95526a25b5 – 1879200a23b7 + 24312150a21b9 + 162081000a19b11 + 6928962750a17b13 – 311803323750a13b17 – 328214025000a11b19 – 2215444668750a9b21 + 7705894500000a7b23 – 17627233668750a5b25 – 9632368125000a3b27 – 11210083593750ab29, y = –a20 – 190a18b2 + 855a16b4 – 102600a14b6 + 1000350a12b8 + 10003500a10b10 + 45015750a8b12 – 207765000a6b14 + 77911875a4b16 – 779118750a2b18 – 184528125b20, z = –a12 + 66a10b2 + 1485a8b4 – 5940a6b6 + 66825a4b8 + 133650a2b10 – 91125b12;

  • x = 3a30 + 30a29b + 435a28b2 – 1160a27b3 + 3915a26b4 + 114318a25b5 – 267525a24b6 – 14857425a22b8 – 54927450a21b9 + 25962975a20b10 + 246123000a19b11 – 679589625a18b12 – 5047022250a17b13 – 16723607625a16b14 – 15511752000a15b15 – 76346904375a14b16 + 78271616250a13b17 – 111704574375a12b18 – 351565695000a11b19 + 513778010625a10b20 + 2039831906250a9b21 – 3497966859375a8b22 – 3404014200000a7b23 – 6263819746875a6b24 – 9457764018750a5b25 – 44594296875a4b26 + 5292677625000a3b27 + 6859171078125a2b28 – 1308568968750ab29 + 1556004528125b30, y = –2a20 – 20a19b – 1140a18b3 + 3990a16b4 + 13680a15b5 – 91200a14b6 – 296400a13b7 – 1556100a12b8 – 642200a11b9 + 1580800a10b10 –2223000a9b11 – 50758500a8b12 + 17214000a7b13 – 124488000a6b14 – 124602000a5b15 + 233343750a4b16 + 421087500a3b17 + 126160000a2b18 + 204347500ab19 – 122051250b20, z = a12 – 12a11b – 66a10b2 – 220a9b3 + 495a8b4 – 3960a7b5 + 3300a6b6 + 27720a5b7 + 32175a4b8 + 47300a3b9 – 67650a2b10 + 95700ab11 + 57025b12;

  • x = 10a30 – 390a29b – 870a28b2 – 14500a27b3 + 15660a26b4 – 363312a25b5 – 2871000a24b6 – 626400a23b7 + 28814400a22b8 – 8004000a21b9 – 27853920a20b10 – 701150400a19b11 + 1155777600a18b12 – 4379788800a17b13 – 18614102400a16b14 – 9927521280a15b15 – 74456409600a14b16 + 74456409600a13b17 + 200983641600a12b18 + 381579494400a11b19 + 108772439040a10b20 + 301001625600a9b21 + 1304716032000a8b22 + 26779852800a7b23 – 2317071974400a6b24 + 1085530152960a5b25 + 797590609920a4b26 + 1297945640960a3b27 – 46432665600a2b28 + 722915819520ab29 + 109893615616b30, y = –7a20 + 20a19b – 760a18b2 – 2280a17b3 – 2280a16b4 + 29184a15b5 – 127680a14b6 – 620160a13b7 + 829920a12b8 – 158080a11b9 + 9231872a10b10 + 6007040a9b11 + 24660480a8b12 + 60119040a7b13 – 49320960a6b14 – 73426944a5b15 – 21815040a4b16 + 73835520a3b17 – 143779840a2b18 – 33187840ab19 – 29857792b20, z = –3a12 – 12a11b + 132a10b2 + 1980a8b4 + 3168a7b5 – 3168a6b6 – 12672a5b7 + 39600a4b8 + 10560a3b9 + 61248a2b10 + 26112ab11 – 27072b12;

  • x = 411a30 + 3810a29b + 60465a28b2 + 383380a27b3* + 614655a26b4 + 2801574a25b5 + 13004325a24b6 + 47449800a23b7 + 158028975a22b8 + 41720850a21b9 – 1905502275a20b10 – 6876436500a19b11 – 11686090125a18b12 – 7100048250a17b13 – 12360927375a16b14 – 50413194000a15b15 – 163600509375a14b16 – 355430126250a13b17 – 463427848125a12b18 – 385737772500a11b19 – 487027141875a10b20 – 383429118750a9b21 + 51944709375a8b22 + 181558125000a7b23 + 5074003125a6b24 – 370510706250a5b25 – 520780640625a4b26 – 336891187500a3b27 – 12290109375a2b28 + 17999531250ab29 + 10367496875b30, y = –41a20 – 740a19b – 1710a18b2 – 14820a17b3 – 61845a16b4 + 13680a15b5 + 216600a14b6 + 250800a13b7 – 2556450a12b8 – 10719800a11b9 – 20426900a10b10 – 11115000a9b11 – 17969250a8b12 –29298000a7b13 – 30609000a6b14 – 58938000a5b15 – 60883125a4b16 + 5557500a3b17 + 23631250a2b18 + 21197500ab19 + 624375b20, z = 10a12 – 12a11b – 264a10b2 – 1540a9b3 – 990a8b4 – 3960a7b5 – 10560a6b6 + 3960a5b7 + 24750a4b8 + 47300a3b9 + 6600a2b10 + 7500ab11 + 10150b12;

  • x = 654a30 + 30630a29b + 113970a28b2 – 138620a27b3 + 203580a26b4 + 17200944a25b5 + 82945800a24b6 + 296913600a23b7 + 777988800a22b8 + 1805702400a21b9* + 2981329920a20b10* + 2958278400a19b11 + 7056326400a18b12 + 35038310400a17b13 + 74456409600a16b14 + 79420170240a15b15* – 210229862400a13b17 – 350383104000a12b18 – 263094681600a11b19 – 159823872000a10b20 – 252439756800a9b21 – 368824320000a8b22 – 466963660800a7b23 – 419069952000a6b24 – 349760913408a5b25 – 160101826560a4b26 – 13227786240a3b27 –456130560a2b28 – 12960399360ab29 – 2136997888b30, y = 127a20 – 20a19b – 10260a18b2 – 31920a17b3 – 69540a16b4 – 207936a15b5 – 1130880a14b6 – 2663040a13b7 – 5690880a12b8 – 5058560a11b9 + 7840768a10b10 + 11381760a9b11 – 13278720a8b12 – 33853440a7b13 – 56033280a6b14 – 42491904a5b15 – 22179840a4b16 – 14008320a3b17 – 20234240a2b18 – 5242880ab19 + 1032192b20, z = 19a12 + 60a11b + 528a10b2 + 440a9b3 – 3960a8b4 – 6336a7b5 – 10560a6b6 – 12672a5b7 – 31680a4b8 – 14080a3b9 + 16896a2b10 + 7680ab11 + 5632b12;

  • x = 7792a30 + 72750a29b + 335820a28b2 + 1274840a27b3 + 8769600a26b4 + 77280534a25b5 + 393144300a24b6 + 1225238400a23b7 + 2442020400a22b + 1731565350a21b9 – 10804319460a20b10 – 53065319400a19b11 – 131393664000a18b12 – 218493292050a17b13 – 266996031300a16b14 – 251290382400a15b15 – 188467786800a14b16 – 123335536950a13b17 – 124721329500a12b18 – 340467348600a11b19 – 950857911360a10b20 – 1638371476350a9b21 – 1773668591100a8b22 – 1303266542400a7b23 – 750953689200a6b24 – 467651900574a5b25 – 295735106700a4b26 – 132724043640a3b27 – 35070382080a2b28 – 3787993350ab29 + 1202316372b30, y = –393a20 – 2440a19b – 7030a18b2 – 6840a17b3 + 110295a16b4 + 1187424a15b5 + 4247640a14b6 + 8290080a13b7 + 10070190a12b8 + 4801680a11b9 – 16992612a10b10 – 47483280a9b11 – 60621210a8b12 – 50971680a7b13 – 47462760a6b14 – 78452064a5b15 – 81097605a4b16 – 43769160a3b17 – 12881430a2b18 – 2653560ab19 – 1426653b20, z = 7a12 + 264a11b + 858a10b2 + 1320a9b3 + 1485a8b4 + 4752a7b5 + 27324a6b6 + 42768a5b7 + 31185a4b8 + 11880a3b9 – 1782a2b10 – 14904ab11 – 4293b12;

  • x = 3125a30 – 1812500a27b3 + 19575000a24b6 – 800400000a21b9 – 12166080000a18b12 – 155725824000a12b18 + 131137536000a9b21 + 41051750400a6b24 + 48653926400a3b27 + 1073741824b30, y = 2500a19b – 142500a16b4 – 912000a13b7 – 15808000a10b10 + 11673600a7b13 –23347200a4b16 – 5242880ab19, z = 25a12 + 2200a9b3 – 10560a6b6 – 28160a3b9 + 4096b12;

  • x = 39151a30 + 578640a29b + 4130760a28b2 + 19156240a27b3 + 59069520a26b4 + 181781280a25b5 + 958809600a24b6 + 5076345600a23b7 + 19392091200a22b8 + 54568070400a21b9 + 115111607040a20b10 + 164280499200a19b11 + 73969766400a18b12 – 350383104000a17b13 – 1191302553600a16b14 – 2223764766720a15b15 – 2978256384000a14b16 – 3083371315200a13b17 – 2553903513600a12b18 – 1701509529600a11b19 – 940256133120a10b20 – 675358310400a9b21 – 944190259200a8b22 – 1252078387200a7b23 – 1137475584000a6b24 – 715121491968a5b25 – 340729528320a4b26 – 134406471680a3b27 – 41964011520a2b28 – 8556380160ab29 – 587202560b30, y = –1153a20 – 11360a19b – 53200a18b2 – 155040a17b3 – 392160a16b4 + 25536a15b5 + 5399040a14b6 + 22617600a13b7 + 51692160a12b8 + 80304640a11b9 + 84351488a10b10 + 40468480a9b11 – 37939200a8b12 – 96890880a7b13 – 95723520a6b14 – 71909376a5b15 – 61870080a4b16 – 46694400a3b17 – 21790720a2b18 – 5242880ab19 – 851968b20, z = –6a12 + 372a11b + 2112a10b2 + 5280a9b3 + 7920a8b4 + 6336a7b5 + 19008a6b6 + 50688a5b7 + 63360a4b8 + 42240a3b9 + 16896a2b10 – 3072ab11 – 3072b12;

  • x = 491520a29b + 7127040a28b2 + 45137920a27b3 + 160358400a26b4 + 410517504a25b5 + 1309593600a24b6 + 6253977600a23b7 + 27661824000a22b8 + 95586969600a21b9 + 261896002560a20b10 + 584432870400a19b11 + 1072074969600a18b12 + 1638041011200a17b13 + 2196464083200a16b14 + 2849198607360a15b15 + 3797276889600a14b16 + 4838571676800a13b17 + 5102636443200a12b18 + 3814863278400a11b19 + 1413894754080a10b20 – 699389520000a9b21 –1600877638800a8b22 – 1501115608800a7b23 – 1055815261200a6b24 – 651072611880a5b25 – 359175395340a4b26 – 159392955780a3b27 – 48132773490a2b28 – 8829176700ab29 – 888155653b30, y = 1024a20 + 10240a19b – 291840a17b3 – 1605120a16b4 – 4902912a15b5 – 11965440a14b6 – 27724800a13b7 – 61176960a12b8 – 117927680a11b9 – 173698304a10b10 – 172149120a9b11 – 104332800a8b12 – 39982080a7b13 – 38139840a6b14 – 70300608a5b15 – 75699420a4b16 – 47230200a3b17 – 20503280a2b18 – 6621140ab19 – 894906b20, z = 64a12 + 384a11b + 2112a10b2 + 7040a9b3 + 7920a8b4 – 6336a7b5 – 32736a6b6 – 50688a5b7 – 61380a4b8 – 57640a3b9 – 24684a2b10 – 1464ab11 + 2353b12;

  • x = 98304a30 + 983040a29b + 7127040a28b2 + 54640640a27b3 + 352788480a26b4 + 1571512320a25b5 + 4436582400a24b6 + 6253977600a23b7 – 8298547200a22b8 – 78682521600a21b9 – 259068349440a20b10 – 559537228800a19b11 – 884230694400a18b12 – 1073048256000a17b13 – 1042389734400a16b14 – 883549393920a15b15 – 837634608000a14b16 – 1189112659200a13b17 – 2085326942400a12b18 – 3274823793600a11b19 – 4055003448480a10b20 – 3830719202400a9b21 – 2794550176800a8b22 – 1667196799200a7b23 – 895510575000a6b24 – 451671056928a5b25 – 191974420980a4b26 – 58334973780a3b27 – 10541110530a2b28 – 501967410ab29 + 184589480b30, y = –2048a20 – 10240a19b + 291840a17b3 + 2115840a16b4 + 8054784a15b5 + 17802240a14b6 + 23347200a13b7 + 11381760a12b8 – 28138240a11b9 – 86058752a10b10 – 126622080a9b11 – 131008800a8b12 – 124725120a7b13 – 129741120a6b14 – 121908864a5b15 – 78137880a4b16 – 31156200a3b17 – 7981520a2b18 – 1769020ab19 – 442143b20, z = 64a12 + 768a11b + 2112a10b2 + 3520a9b3 + 7920a8b4 + 25344a7b5 + 55968a6b6 + 60192a5b7 + 33660a4b8 + 6160a3b9 – 8844a2b10 – 10308ab11 – 2207b12;

  • x = 81250a30 + 656250a29b + 2718750a28b2 + 10875000a27b3 + 73406250a26b4 + 484481250a25b5 + 1936293750a24b6 + 4854600000a23b7 + 7316156250a22b8 + 162581250a21b9 – 39972476250a20b10 – 135277605000a19b11 – 265610238750a18b12 – 363128973750a17b13 – 370827821250a16b14 – 294723288000a15b15 – 190503704250a14b16 – 121042991250a13b17 – 148321623750a12b18 – 341384607000a11b19 – 675919490850a10b20 – 884003881050a9b21 – 767542679550a8b22 – 472107657600a7b23 – 239161464450a6b24 – 128632685418a5b25 – 66582538110a4b26 – 24423970520a3b27 – 5299948650a2b28 – 427589970ab29 + 128301258b30, y = –1875a20 – 10000a19b – 23750a18b2 + 463125a16b4 + 3192000a15b5 + 9063000a14b6 + 14592000a13b7 + 14264250a12b8 + 2964000a11b9 – 23687300a10b10 – 49004800a9b11 – 51684750a8b12 – 38760000a7b13 –34804200a6b14 – 45089280a5b15 – 37078215a4b16 – 16548240a3b17 – 4149030a2b18 – 789760ab19 – 322343b20, z = 25a12+ 600a11b + 1650a10b2 + 2200a9b3 + 2475a8b4 + 7920a7b5 + 31020a6b6 + 39600a5b7 + 24255a4b8 + 7480a3b9 – 2046a2b10 – 7368ab11 – 1763b12;

  • x = 81250a30 + 656250a29b + 2718750a28b2 + 10875000a27b3 + 73406250a26b4 + 484481250a25b5 + 1936293750a24b6 + 4854600000a23b7 + 7316156250a22b8 + 162581250a21b9 – 39972476250a20b10 – 135277605000a19b11 – 265610238750a18b12 – 363128973750a17b13 – 370827821250a16b14 – 294723288000a15b15 – 190503704250a14b16 – 121042991250a13b17 – 148321623750a12b18 – 341384607000a11b19 – 675919490850a10b20 – 884003881050a9b21 – 767542679550a8b22 – 472107657600a7b23 – 239161464450a6b24 – 128632685418a5b25 – 66582538110a4b26 – 24423970520a3b27 – 5299948650a2b28 – 427589970ab29 + 128301258b30, y = –1875a20 – 10000a19b – 23750a18b2 + 463125a16b4 + 3192000a15b5 + 9063000a14b6* + 14592000a13b7 + 14264250a12b8 + 2964000a11b9 – 23687300a10b10 – 49004800a9b11 – 51684750a8b12 – 38760000a7b13 –34804200a6b14 – 45089280a5b15 – 37078215a4b16 – 16548240a3b17 – 4149030a2b18 – 789760ab19 – 322343b20, z = 25a12 + 600a11b + 1650a10b2 + 2200a9b3 + 2475a8b4 + 7920a7b5 + 31020a6b6 + 39600a5b7 + 24255a4b8 + 7480a3b9 – 2046a2b10 – 7368ab11 – 1763b12;

  • x = 19683a30 – 590490a29b – 8562105a28b2 – 45664560a27b3 – 111307365a26b4 – 154117890a25b5 – 442375425a24b6 – 2968196400a23b7 – 13456775025a22b8 – 42084331650a21b9 – 102118323645a20b10 – 203055076800a19b11 – 323582560425a18b12 – 408808802250a17b13 – 459390230325a16b14 – 577967879520a15b15 – 844178628375a14b16 – 1119258449550a13b17 – 1068882324075a12b18 – 561383751600a11b19 + 62314471665a10b20 + 395258830650a9b21 + 394892797725a8b22 + 275540518800a7b23 + 175903201125a6b24 + 112283394858a5b25 + 67302682785a4b26 + 30539055520a3b27 + 8170447965a2b28 + 1348449810ab29 + 168454745b30, y = 1458a20 + 14580a19b – 277020a17b – 1246590a16b4 – 2881008a15b5 – 5909760a14b6 – 12927600a13b7 – 27609660a12b8 – 52551720a11b9 – 71278272a10b10 – 53618760a9b11 – 15338700a8b12 + 3324240a7b13 – 10177920a6b14 – 28544688a5b – 28175670a4b16 – 14381100a3b17 – 6013120a2b18 – 2236540ab19 – 251382b20, z = 81a12 + 324a11b + 1782a10b2 + 5940a9b3 + 4455a8b4 – 7128a7b5 – 22572a6b6 – 26136a5b7 – 31185a4b8 – 31020a3b9 – 9834a2b10 + 804ab11 + 1657b12;

  • x = 2343750a29b – 18125000a27b3 + 298518750a25b5 + 1174500000a23b7 + 3039018750a21b9 – 4052025000a19b11 + 34644813750a17b13 – 62360664750a17b17 + 13128561000a11b19 – 17723557350a9b21 – 12329431200a7b23 – 5640714774a5b25 + 616471560a3b27 – 143489070ab29, y = 3125a20 – 118750a18b2 – 106875a16b4 – 2565000a14b6 – 5001750a12b8 + 10003500a10b10 – 9003150a8b12 – 8310600a6b14 – 623295a4b16 – 1246590a2b18 + 59049b20, z = 125a12 + 1650a10b2 – 7425a8b4 – 5940a6b6 – 13365a4b8 + 5346a2b10 + 729b12;

  • x = 2343750a29b + 18125000a27b3 + 298518750a25b5 – 1174500000a23b7 + 3039018750a21b9 + 4052025000a19b11 + 34644813750a17b13 – 62360664750a17b17 – 13128561000a11b19 – 17723557350a9b21 + 12329431200a7b23 – 5640714774a5b25 – 616471560a3b27 – 143489070ab29, y = –3125a20 – 118750a18b2 + 106875a16b4 – 2565000a14b6 + 5001750a12b8 + 10003500a10b10 + 9003150a8b12 – 8310600a6b14 + 623295a4b16 – 1246590a2b18 – 59049b20, z = –125a12 + 1650a10b2 + 7425a8b4 – 5940a6b6 + 13365a4b8 + 5346a2b10 – 729b12;

  • x = 334611a30 + 1771470a29b + 8562105a28b2 + 102745260a27b3 + 402418935a26b4 + 880184394a25b5 + 1641070125a24b6 + 2854035000a23b7 + 4923210375a22b8 + 12763878750a21b9 + 27022954725a20b10 + 31070927700a19b11 – 15936614325a18b12 – 101162856150a17b13 – 153130076775a16b14 – 138209710320a15b15 – 90307481175a14b16 – 67903834950a13b17 – 127210623525a12b18 – 179245377900a11b19 – 156621301515a10b20 – 113032588050a9b21 – 70044654825a8b22 – 31858547400a7b23 – 7215286275a6b24 + 490350618a5b25 + 177854535a4b26 – 1342327060a3b27 – 657823095a2b28 – 103017570ab29 + 4942947b30, y = –729a20 – 43740a19b – 138510a18b2 – 277020a17b3 – 1315845a16b4 – 2881008a15b5 – 3139560a14b6 + 369360a13b7 + 4201470a12b8 + 1867320a11b9 – 18166356a10b10 – 30143880a9b11 – 23541570a8b12 – 16046640a7b13 – 9377640a6b14 – 4807152a5b15 – 5447205a4b16 – 2876220a3b17 – 570190a2b18 + 280260ab19 + 53831b20, z = 162a12 + 324a11b – 5940a9b3 – 8910a8b4 – 7128a7b5 – 14256a6b6 – 11880a5b7 – 2970a4b8 + 11220a3b9 + 5808a2b10 + 1116ab11 + 710b12;

  • x = 531441a30 – 493177248a25b5 – 16804558080a20b10 – 53110702080a10b20 + 4926210048a5b25 + 16777216b30, y = –6561a20 – 2659392a15b5 – 10243584a10b10 + 8404992a5b15 – 65536b20, z = –972a11b + 19008a6b6 + 3072ab11;

  • x = 647992a30 + 6865710a29b + 60148320a28b2 + 327777720a27b3 + 1008911160a26b4 + 2330215830a25b5 + 5138568000a24b6 + 11560212000a23b7 + 24474231000a22b8 + 37074027750a21b9 + 11987750880a20b10 – 100080815400a19b11 – 281880469800a18b12 – 418321156050a17b13 – 437431406400a16b14 – 414474013440a15b15 – 475822992600a14b16 – 632485984950a13b17 – 739119775200a12b18 – 669407736600a11b19 – 482938228440a10b20 – 287513985150a9b21 – 131883508800a8b22 – 40068928800a7b23 – 7215449400a6b24 – 4046295006a5b25 – 5076470880a4b26 – 3161731960a3b27 – 847439160a2b28 – 79084230ab29 + 5574720b30, y = –5879a20 – 77680a19b – 282910a18b2 – 962160a17b3 – 2793855a16b4 – 4009152a15b5 – 1780680a14b6 + 1842240a13b7 – 4972110a12b8 – 30786080a11b9 – 60768916a10b10 – 63350560a9b11 – 46734870a8b12 – 31427520a7b13 – 20695560a6b14 – 14923968a5b15 – 9746715a4b16 – 3114480a3b17 + 42370a2b18 + 377360ab19 + 51701b20, z = 185a12 + 144a11b – 2046a10b2 – 9680a9b3 – 13365a8b4 – 15840a7b5 – 20724a6b6 – 9504a5b7 + 8415a4b8 + 16720a3b9 + 6930a2b10 + 1776ab11 + 701b12;

  • x = 1090625a30 + 7031250a29b + 17671875a28b2 – 34437500a27b3 – 648421875a26b4 – 3636056250a25b5 – 11556590625a24b6 – 25545375000a23b7 – 49468471875a22b8 – 105690318750a21b9 – 234308345625a20b10 – 428299042500a19b11 – 589603404375a18b12 – 612058376250a17b13 – 490801528125a16b14 – 314112978000a15b15 – 176688550125a14b16 – 131650292250a13b17 – 197475438375a12b18 – 347906866500a11b19 – 472956410025a10b20 – 443380679550a9b21 – 281935847475a8b22 – 116558789400a7b23 – 21219750225a6b24 + 9832721382a5b25 + 9084393405a4b26 + 3230767620a3b27 + 642157875a2b28 + 92706930ab29 + 16277841b30, y = –10625a20 – 47500a19b – 118750a18b2 – 427500a17b3 – 2458125a16b4 – 9918000a15b5 – 21033000a14b6 – 27702000a13b7 – 28343250a12b8 – 33345000a11b9 – 48683700a10b10 – 54685800a9b11 – 39013650a8b12 – 12927600a7b13 + 10157400a6b14 + 21053520a5b15 + 14751315a4b16 + 5263380a3b17 + 969570a2b18 + 14580ab19 – 53217b20, z = –100a12 – 1500a11b – 3300a10b2 – 3300a9b3 + 11880a7b5 + 35640a6b6 + 35640a5b7 + 17820a4b8 + 5940a3b9 + 3564a2b10 + 3564ab11 + 648b12;

  • x = 327680a30 – 6389760a29b – 7127040a28b2 – 59392000a27b3 + 32071680a26b4 – 372031488a25b5 – 1469952000a24b6 – 160358400a23b7 + 3688243200a22b8 – 512256000a21b9 – 891325440a20b10 – 11218406400a19b11 + 9246220800a18b12 – 17519155200a17b13 – 37228204800a16b14 – 9927521280a15b15 – 37228204800a14b16 + 18614102400a13b17 + 25122955200a12b18 + 23848718400a11b19 + 3399138720a10b20 + 4703150400a9b21 + 10193094000a8b22 + 104608800a7b23 – 4525531200a6b24 + 1060088040a5b25 + 389448540a4b26 + 316881260a3b27 – 5668050a2b28 + 44123280ab29 + 3353687b30, y = –7168a20 + 10240a19b – 194560a18b2 – 291840a17b3 – 145920a16b4 + 933888a15b5 – 2042880a14b6 – 4961280a13b7 + 3319680a12b8 – 316160a11b9 + 9231872a10b10 + 3003520a9b11 + 6165120a8b12 + 7514880a7b13 – 3082560a6b14 – 2294592a5b15 – 340860a4b16 + 576840a3b17 – 561640a2b18 – 64820ab19 – 29158b20, z = –192a12 – 384a11b + 2112a10b2 + 7920a8b4 + 6336a7b5 – 3168a6b6 – 6336a5b7 + 9900a4b8 + 1320a2b9 + 3828a2b10 + 816ab11 – 423b12;

  • x = 2730128a30 + 44424330a29b + 222407670a28b2 + 660279540a27b3 + 2555226540a26b4 + 12752601984a25b5 + 49309216200a24b6 + 139926484800a23b7 + 306268257600a22b8 + 539008569600a21b9 + 784750579200a20b10 + 992175840000a19b11 + 1208091744000a18b12 + 1532926080000a17b13 + 1861410240000a16b14 + 1826663915520a15b15 + 1191302553600a14b16 + 210229862400a13b17 – 552826675200a12b18 – 784366387200a11b19 – 637820190720a10b20 – 413083238400a9b21 – 250800537600a8b22 – 148812595200a7b23 – 80393011200a6b24 – 36125540352a5b25 – 12315525120a4b26 – 2888826880a3b27 – 456130560a2b28 – 62914560ab29 – 6291456b30, y = 12931a20 – 38060a19b – 852340a18b2 – 3456480a17b3 – 8569380a16b4 – 18666816a15b5 – 39982080a14b6 – 71829120a13b7 – 97693440a12b8 – 89789440a11b9 – 45021184a10b10 – 8852480a9b11 – 13278720a8b12 – 33853440a7b13 – 39690240a6b14 – 27549696a5b15 – 12840960a4b16 – 4669440a3b17 – 1556480a2b18 – 327680ab19 – 16384b20, z = 395a12 + 1836a11b + 6072a10b2 + 5720a9b3 – 11880a8b4 – 31680a7b5 – 40128a6b6 – 38016a5b7 – 31680a4b8 – 14080a3b9 + 1536ab11 + 512b12;

  • x = 4387861a30 + 1976910a29b – 450850965a28b2 – 4115300680a27b3 – 18483783795a26b4 – 52053249618a25b5 – 102595034925a24b6 – 163810490400a23b7 – 288536145975a22b8 – 675364413450a21b9 – 1649574184905a20b10 – 3469944105000a19b11 – 6043269274575a18b12 – 8709877262250a17b13 – 10416189942225a16b14 – 10430102044800a15b15 – 8985230820225a14b16 – 7025882685750a13b17 – 5285702680575a12b18 – 3824469279000a11b19 – 2458531261305a10b20 – 1260330950550a9b21 – 452454063975a8b22 – 77491317600a7b23 + 23954991075a6b + 25048268658a5b25 + 11603147805a4b26 + 3900051080a3b27 + 974522235a2b28 + 156118290ab29 + 12092581b30, y = 29396a20 + 387860a19b + 1429940a18b2 + 1075020a17b3 – 8603580a15b4 – 35858928a15b5 – 78801360a14b6 – 128815440a13b7 – 182078520a12b8 – 242642920a11b9 – 300571336a10b10 – 308265880a9b11 – 235430520a8b12 – 125450160a7b13 – 45969360a6b14 – 14994192a5b15 – 8193180a4b16 – 4814220a3b17 – 1853260a2b18 – 453460ab19 – 52684b20, z = 537a12 + 2460a11b + 8778a10b2 + 27060a9b3 + 44055a8b4 + 32472a7b5 – 5940a6b6 – 32472a5b7 – 35145a4b8 – 27060a3b9 – 12342a2b10 – 2460ab11 + 57b12;

  • x = 1525899a30 – 16253970a29b – 183283335a28b2 – 976109840a27b3 – 3225987405a26b4 – 8310854394a25b5 – 22076391375a24b6 – 56106334800a23b7 – 106221134025a22b8 – 135194463450a21b9 – 108742574115a20b10 – 59127148800a19b11 – 50324799825a18b12 – 103934441250a17b13 – 231658321275a16b14 – 427193650080a15b15 – 624299543775a14b16 – 699825237750a13b17 – 570253634325a12b18 – 299331190800a11b19 – 55336884615a10b20 + 59005588050a9b21 + 71878871475a8b22 + 47719465200a7b23 + 21048508125a6b24 + 5798257506a5b25 + 1018890495a4b26 + 273987360a3b27 + 111307365a2b28 + 33657930ab29 + 4979799b30, y = –20240a20 – 73980a19b – 562780a18b2 – 2664180a17b3 – 6887880a16b4 – 9699120a15b5 – 10218960a14b6 – 18426960a13b7 – 30944160a12b8 – 32277960a11b9 – 12004200a10b10 + 16272360a9b11 + 33611760a8b12 + 37305360a7b13 + 26224560a6b14 + 9972720a5b15 + 1108080a4b16 – 277020a3b17 – 277020a2b18 – 131220ab19 – 29160b20, z = – 359a12 – 1692a11b + 66a10b2 + 4620a9b3 + 16335a8b4 + 30888a7b5 + 27324a6b6 + 7128a5b7 + 4455a14b8 + 5940a3b9 + 5346a2b10 + 2268ab11 + 81b12;

  • x = 1506463a30 + 25129050a29b + 80181375a28b2 + 295423000a27b3 + 908299575a26b4 + 2205784602a25b5 + 969060375a24b6 – 3460860000a23b7 + 33766875a22b8 + 15646319250a21b9 + 32981112315a20b10 + 58931451000a19b11 + 100774111875a18b12 + 134729831250a17b13 + 90889171875a16b14 – 90889171875a14b16 – 134729831250a13b17 – 100774111875a12b18 – 58931451000a11b19 – 32981112315a10b20 – 15646319250a9b21 – 33766875a8b22 + 3460860000a7b23 – 969060375a6b24 – 2205784602a5b25 – 908299575a4b26 – 295423000a3b27 – 80181375a2b28 – 25129050ab29 – 1506463b30, y = –16514a20 – 83100a19b – 494000a18b2 – 541500a17b3 – 401850a16b4 – 2155056a15b5 – 5836800a14b6 – 342000a13b7 + 11485500a12b8 + 17438200a11b9 + 22360416a10b10 + 17438200a9b11 + 11485500a8b12 – 342000a7b13 – 5836800a6b14 – 2155056a5b15 – 401850a4b16 – 541500a3b17 – 494000a2b18 – 83100ab19 – 16514b20, z = –295a12 + 204a11b + 3630a10b2 + 5500a9b3 + 12375a8b4 + 3960a7b5 – 4092a6b6 + 3960a5b7 + 12375a4b8 + 5500a3b9 + 3630a2b10 + 204ab11 – 295b12.

 

Da notare quindi che in tutte le soluzioni note (escluse quelle banali) almeno un esponente è 2.

In effetti il milionario americano Andrew Beal propose nel 1997 la “congettura di Beal” (nota anche come “congettura di Tijdeman – Zagier”, perché proposta dai due matematici nel 1994), ossia che in tutte le soluzioni almeno uno degli esponenti sia 2, offrendo un premio di 100000 dollari per chi fosse riuscito a dimostrarla o a trovare un controesempio, premio nel frattempo salito a un milione di dollari. L’unico caso ancora aperto è quello iperbolico, nel quale non è certo che tutte le soluzioni siano state trovate.

 

La congettura è una conseguenza della congettura “abc”, che sfortunatamente appare ancora più difficile da dimostrare

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