Solutions of Diophantine equation x/y+y/z+z/x=n (-1 >= n >= -100) When (x,y,z) is a solution of the equation and l=lcm(x,y,z), then (l/z,l/y,l/x) also becomes the solution. Parametrized solutions for -n^2. (x,y,z) : (1,n,-n^2), (-1,n,n^2) n | (x, y, z) | lcm(x,y,z) | ----+--------------------------------------------------------------------------- -4 | (-1, 2, 4) | 4 | Parametrized solution -4 | (1, 2, -4) | 4 | Parametrized solution -4 | (112, 441, -324) | 63504 | -4 | (144, 567, -196) | 63504 | derived from (112, 441, -324) -9 | (-1, 3, 9) | 9 | Parametrized solution -9 | (1, 3, -9) | 9 | Parametrized solution ----+--------------------------------------------------------------------------- -10 | (-4, 112, 49) | 784 | -10 | (7, -196, 16) | 784 | derived from (-4, 112, 49) -11 | (144, 1539, -1444) | 467856 | -11 | (304, 3249, -324) | 467856 | derived from (144, 1539, -1444) -12 | (126, 3724, -1083) | 636804 | -12 | (171, 5054, -588) | 636804 | derived from (126, 3724, -1083) -16 | (-1, 4, 16) | 16 | Parametrized solution -16 | (1, 4, -16) | 16 | Parametrized solution -16 | (-20, 325, 338) | 16900 | -16 | (50, 52, -845) | 16900 | derived from (-20, 325, 338) -17 | (7, 441, -81) | 3969 | -17 | (9, 567, -49) | 3969 | derived from (7, 441, -81) ----+--------------------------------------------------------------------------- -21 | (-1813, 106782, 42588) | 408120804 | -21 | (3822, -225108, 9583) | 408120804 | derived from (-1813, 106782, 42588) -22 | (4, 144, -81) | 1296 | -22 | (9, 324, -16) | 1296 | derived from (4, 144, -81) -24 | (-2, 28, 49) | 196 | -24 | (4, 7, -98) | 196 | derived from (-2, 28, 49) -25 | (-1, 5, 25) | 25 | Parametrized solution -25 | (1, 5, -25) | 25 | Parametrized solution -25 | (2, 28, -49) | 196 | -25 | (-4, 7, 98) | 196 | derived from (2, 28, -49) -27 | (85456, 3314799, -2179548) | 725065874064 | -27 | (218736, 8484669, -332668) | 725065874064 | derived from (85456, 3314799, -2179548) -28 | (38236250, 237582772, -1068264925) | 45496655370302500 | -28 | (-42589300, 191498125, 1189882778) | 45496655370302500 | derived from (38236250, 237582772, -1068264925) -29 | (5994, 695452, -145161) | 7154114724 | -29 | (10287, 1193546, -49284) | 7154114724 | derived from (5994, 695452, -145161) ----+--------------------------------------------------------------------------- -32 | (2097490251143664, 149657797996601921, -62093731192983612) | 724265902083081258793411970774544 | -32 | (4839479878619664, 345301200655484671, -11664074426967612) | 724265902083081258793411970774544 | derived from (2097490251143664, 149657797996601921, -62093731192983612) -33 | (117, 5915, -3675) | 1863225 | -33 | (315, 15925, -507) | 1863225 | derived from (117, 5915, -3675) -34 | (-112, 1519, 3844) | 753424 | -34 | (196, 496, -6727) | 753424 | derived from (-112, 1519, 3844) -35 | (25267361580, 4697907757200, -719516461237) | 193941199109727589136400 | -35 | (41282462137, 7675561949580, -269543797200) | 193941199109727589136400 | derived from (25267361580, 4697907757200, -719516461237) -36 | (-1, 6, 36) | 36 | Parametrized solution -36 | (1, 6, -36) | 36 | Parametrized solution -37 | (-18772, 521872, 707731) | 36360387856 | -37 | (51376, 69673, -1936948) | 36360387856 | derived from (-18772, 521872, 707731) -38 | (-1262205247106475, 4838366738762777500, 105726065143878576) | 7470343230209374294987772170410000 | -38 | (1543980362290524, -5918485323472279600, 70657535774581875) | 7470343230209374294987772170410000 | derived from (-1262205247106475, 4838366738762777500, 105726065143878576) ----+--------------------------------------------------------------------------- -40 | (126763588, 31487624080, -4096748425) | 6442421573167056400 | first -40 | (204601705, 50822335300, -1572569488) | 6442421573167056400 | derived from (126763588, 31487624080, -4096748425) -44 | (24187, 1050426, -1040364) | 88733685924 | -44 | (84474, 3668652, -85291) | 88733685924 | derived from (24187, 1050426, -1040364) -45 | (-22932, 602992, 1044309) | 59300042256 | -45 | (56784, 98343, -2585908) | 59300042256 | derived from (-22932, 602992, 1044309) -46 | (-200, 2752, 9245) | 2958400 | -46 | (320, 1075, -14792) | 2958400 | derived from (-200, 2752, 9245) -47 | (15876, 8489936, -439569) | 151978344336 | -47 | (17901, 9572836, -345744) | 151978344336 | derived from (15876, 8489936, -439569) -48 | (72072752816411426700, 33132848506525529596688, -2507202774146263930905) | 3297207126012183947487833614054532620256400 | -48 | (99514749701125078425, 45748316765575087070092, -1315093920608366852880) | 3297207126012183947487833614054532620256400 | derived from (72072752816411426700, 33132848506525529596688, -2507202774146263930905) -49 | (-1, 7, 49) | 49 | Parametrized solution -49 | (1, 7, -49) | 49 | Parametrized solution -49 | (7, 882, -324) | 15876 | -49 | (18, 2268, -49) | 15876 | derived from (7, 882, -324) -49 | (-7, 931, 361) | 17689 | -49 | (19, -2527, 49) | 17689 | derived from (-7, 931, 361) ----+--------------------------------------------------------------------------- -50 | (-23228576324023444416, 854759290074299928576, 1177657176840826184167) | 81769285649336660559271078578077874130944 | -50 | (69433861787086712832, 95663523753253231769, -3520202207346184987584) | 81769285649336660559271078578077874130944 | derived from (-23228576324023444416, 854759290074299928576, 1177657176840826184167) -55 | (3724, 941488, -185983) | 7519664656 | -55 | (7987, 2019244, -40432) | 7519664656 | derived from (3724, 941488, -185983) -57 | (3, 117, -169) | 1521 | -57 | (-9, 13, 507) | 1521 | derived from (3, 117, -169) -59 | (-4167801316, 691599901104, 257087122857) | 8188949054796871357584 | -59 | (11840587371, -1964812723524, 31852816912) | 8188949054796871357584 | derived from (-4167801316, 691599901104, 257087122857) ----+--------------------------------------------------------------------------- -60 | (-12, 171, 722) | -12996 | -60 | (18, 76, -1083) | -12996 | derived from (-12, 171, 722) -60 | (2925, 2162862, -124820) | 8543304900 | -60 | (3950, 2920788, -68445) | 8543304900 | derived from (2925, 2162862, -124820) -60 | (-11530939, 121550772, 692785584) | 9805750199424144 | -60 | (14154091, 80672052, -850386096) | 9805750199424144 | derived from (-11530939, 121550772, 692785584) -64 | (-1, 8, 64) | 64 | Parametrized solution -64 | (1, 8, -64) | 64 | Parametrized solution -65 | (514525483521439325, 156371448711883658877, -30836716397436136365) | 183268707585397474444279056443471733225 | -65 | (1172008759240136925, 356189758243064760573, -5943197882140104365) | 183268707585397474444279056443471733225 | derived from (514525483521439325, 156371448711883658877, -30836716397436136365) -66 | (55209567, 51929429328, -2494613248) | 3712109201277974784 | -66 | (71483728, 67236701952, -1488049983) | 3712109201277974784 | derived from (55209567, 51929429328, -2494613248) -67 | (8925639164554048, 12295050524054779, -604315830321469864) | 1638409319290756838015440596134464 | -67 | (-2711180540180776, 133257632092302016, 183562128054346093) | 1638409319290756838015440596134464 | derived from (-2711180540180776, 133257632092302016, 183562128054346093) -68 | (1119650, 1045080396, -54775035) | 1601559579860100 | -68 | (1532475, 1430410914, -29238860) | 1601559579860100 | derived from (1119650, 1045080396, -54775035) ----+--------------------------------------------------------------------------- -72 | (-684, 32851, 49686) | 107619876 | -72 | (2166, 3276, -157339) | 107619876 | derived from (-684, 32851, 49686) -72 | (156663, 5686424, -11204544) | 4635891284544 | -72 | (-413751, 815256, 29591488) | 4635891284544 | derived from (156663, 5686424, -11204544) -73 | (6892846675, 4467884509753, -431880684235) | 56136184591117079674225 | -73 | (12564376825, 8144122049707, -129980771635) | 56136184591117079674225 | derived from (6892846675, 4467884509753, -431880684235) -76 | (-490, 7300, 37303) | 26112100 | -76 | (700, 3577, -53290) | 26112100 | derived from (-490, 7300, 37303) -77 | (-1741895946, 72104680116, 135014171533) | 660049762581289384164 | -77 | (4888744308, 9154048829, -378926057034) | 660049762581289384164 | derived from (-1741895946, 72104680116, 135014171533) second -79 | (-1521727, 1194808356, 133802928) | 3897029947030416 | -79 | (3261636, -2560925808, 29125147) | 3897029947030416 | derived from (-1521727, 1194808356, 133802928) ----+--------------------------------------------------------------------------- -81 | (-1, 9, 81) | 81 | Parametrized solution -81 | (1, 9, -81) | 81 | Parametrized solution -82 | (202806672320, 334397609743025, -9465528335816) | 74414946812532698449806400 | -82 | (222534326336, 366925535344895, -7861679155400) | 74414946812532698449806400 | derived from (202806672320, 334397609743025, -9465528335816) -84 | (218736, 61727913, -17607772) | 38376984247056 | -84 | (621712, 175448871, -2179548) | 38376984247056 | derived from (218736, 61727913, -17607772) -85 | (724774810, 760312837700, -50747851843) | 921316222701459516100 | -85 | (1211759393, 1271175832810, -18154782700) | 921316222701459516100 | derived from (724774810, 760312837700, -50747851843) -88 | (-1264, 1092175, 122500) | 3058090000 | -88 | (2800, -2419375, 24964) | 3058090000 | derived from (-1264, 1092175, 122500) -88 | (30376816105, 15519532104200, -2483383961152) | 1110818656125969595662400 | -88 | (71575524872, 36567975138880, -447300406825) | 1110818656125969595662400 | derived from (30376816105, 15519532104200, -2483383961152) -89 | (5239, 181629, -464373) | 5801411889 | -89 | (-12493, 31941, 1107351) | 5801411889 | derived from (5239, 181629, -464373) ----+--------------------------------------------------------------------------- -90 | (9, 2268, -784) | 63504 | -90 | (28, 7056, -81) | 63504 | derived from (9, 2268, -784) -92 | (-2352276, 7167484341, 277225238) | 27954937796079204 | -92 | (3900244, -11884208229, 100838358) | 27954937796079204 | derived from (-2352276, 7167484341, 277225238) -92 | (-261866739330, 54219465971100, 24666094355161) | 49684849256633076697824900 | -92 | (916365522359, -189733333006530, 2014297380900) | 49684849256633076697824900 | derived from (-261866739330, 54219465971100, 24666094355161) -94 | (-704, 11011, 66248) | 64128064 | -94 | (968, 5824, -91091) | 64128064 | derived from (-704, 11011, 66248) -95 | (18392, 8941248, -1647459) | 418691820096 | -95 | (46827, 22764888, -254144) | 418691820096 | derived from (18392, 8941248, -1647459) -95 | (889380, 1975106800, -47545083) | 1910684741504400 | -95 | (967383, 2148333380, -40186800) | 1910684741504400 | derived from (889380, 1975106800, -47545083) -100 | (-1, 10, 100) | 100 | Parametrized solution -100 | (1, 10, -100) | 100 | Parametrized solution -----+--------------------------------------------------------------------------