x, y, z, w in N (integer > 0)
m=x2, n=gy2
X=mg/n, Y=kg2/n2
We are using the same characters x, y, z in the definition 2 and 3,
but I think there's no confusion.
There are 361 congruent numbers in the range of 1 <= g <= 999.
g, m, n 5, 5, 4 6, 2, 1 7, 16, 9 13, 325, 36 14, 8, 1 15, 4, 1 21, 4, 3 22, 50, 49 23, 24336, 17689 29, 4901, 4900 30, 3, 2 31, 1600, 81 34, 9, 8 37, 777925, 1764 38, 1250, 289 39, 13, 12, 41, 25, 16, 46, 72, 49, 47, 14561856, 2289169, 53, 1873180325, 1158313156, 55, 125, 44, 61, 12079525, 10227204 62, 39200, 22801, 65, 9, 4, 69, 192, 169, 70, 7, 2, 71, 3600, 121, 77, 2816, 2809, 78, 26, 1, 79, 169000000, 166952241, 85, 85, 36, 86, 338, 49 87, 17956, 169 93, 1444, 75 94, 14112, 529 95, 1445, 76 101, 44715091781, 3975302500 102, 50, 1 103, 8780605285453456, 7551929273974569 109, 2725, 1764 110, 10, 1 111, 37, 12 118, 716311250, 19298449 119, 144, 25 127, 306317326339867638016, 305111826865145547009 133, 256036, 143811 134, 2084882, 14161 137, 3425, 3136 138, 24, 1 141, 48, 1 142, 4918336200, 164070481 143, 101124, 1849 145, 29, 20 149, 93125, 56644 151, 115600, 35721 154, 9, 2 157, 443624018997429899709925, 166136231668185267540804 158, 768800, 579121 159, 33124, 11449 161, 16, 7 165, 16, 11 166, 197646962, 96020401 167, 115222229136, 3447686089 173, 2404644000341688241925, 2367961190733987384484 174, 27, 2 181, 3073858021, 1221502500 182, 343, 18 183, 17853541, 456300 190, 10, 9 191, 404727588018561600, 384957657745092721 194, 97, 72 197, 1991322221917925, 103880003159716 199, 13933945152400, 27368486201 205, 45, 4 206, 3886472, 1216609 210, 5, 2 213, 7680000, 724201 214, 195938, 82369 215, 3844, 361 219, 73, 48 221, 13, 4 222, 12506, 3481 223, 50942322536708571136, 14384824876061849889 226, 81, 32 229, 874407840421, 7574220900 230, 1127, 722 231, 7, 4 237, 1888250116, 104961675 238, 1800, 1681 239, 14400, 14161 246, 578, 529 247, 35976004, 34539129 253, 125316, 91091 254, 147968, 25921 255, 16, 1 257, 23409, 10816 262, 107898050, 91259809 263, 2415046965407199886472444395015056, 2196589972531420851340521356470969 265, 49, 4 269, 1973658180741549394113125, 6841040196454710370084 271, 90565100276929600, 10384564009711761 277, 14983858080745346212342049973925, 1065773764946955463053657843204 278, 983320808450, 981718327489 285, 76, 49 286, 11, 2 287, 19017513216, 19017237409 291, 49, 48 293, 15705303325706301534145925, 14764613336211576651809284 295, 937445, 226796 299, 36, 13 301, 2752, 729 302, 9398643356699544200, 9025426439116032001 303, 40822586116, 27689958409 309, 11957764, 1915203 310, 40, 9 311, 190727483528715254643600, 124670583817928357228041 313, 169, 144 317, 2761253095084587196325, 283607850589557281956 318, 1083, 242 319, 2500, 2401 323, 425, 304 326, 2266125842, 461041 327, 6922040749, 6897607500 330, 6, 5 334, 3274885399752, 1127604000769 335, 7828804, 6911641 341, 13924, 2475 349, 19385975941, 12934512900 353, 289, 64 357, 21, 4 358, 956307324451250, 419801702617969 359, 3302812394010000, 3302771130607081 365, 80957528885, 9001075876 366, 122, 121 367, 268375455300992413724915588959538171729183296, 246962909981075733725521615532807811435729 371, 53, 28 373, 2528397723062320973179188676827215493925, 268218643743370556175474368869395176964 374, 1682, 1 381, 158404, 146523 382, 33312967200, 7863965041 383, 403654067228736, 303839168346289 386, 121, 72 389, 84049783723809350326254513125, 5237316101673450088070309764 390, 8, 5 391, 327184, 308025 395, 80, 1 397, 4720652373925, 4280901693156 398, 860278445000, 650976262561 399, 64, 57 406, 3712, 9 407, 4662023125, 4661617356 410, 41, 40 413, 4110604996, 2277480771 415, 881971204, 245580241 421, 452568163984303525, 36031587602493924 422, 638450, 47089 426, 96, 25 429, 12, 1 430, 3483, 3362 431, 193276189912334400, 3177076466844721 434, 32, 31 437, 79459396, 29545171 438, 49, 24 439, 70989410337024400, 43044356506452921 442, 121, 104 445, 89, 80 446, 2835884807783552, 377414273685409 447, 503643364, 461433361 453, 1708547008996, 1567254468675 454, 1002780063842, 856940855521 455, 64, 1 457, 64009, 41616 461, 50724269719704638882388101, 36304957961648911912322500 462, 14, 11 463, 1162623700824087616, 141856298295251409 465, 16, 15 469, 1467568, 1212201 470, 47, 2 471, 141394357, 11548332 478, 115925065606428893644500000, 1634220262284462674687761 479, 2145542400, 522076801 485, 19921511285, 6611966596 487, 1019034345734453406615667216, 409136538889385517944610009 493, 59747170703125, 9162402086916 494, 2299, 1458 501, 239341872, 17330569 502, 347052959946050, 33057359699809 503, 15881569125140359780032984660061332598637667216, 15492409457317760025384676078081128557650708009 505, 101, 20 509, 86021, 28900 510, 32, 17 511, 359178304, 4601025 514, 257, 32 517, 36, 11 518, 1184, 841 519, 326025012196, 4124593729 526, 28486152, 18769 527, 576, 49 533, 19348233125, 18912300484 534, 49298, 47089 535, 174724, 96721 541, 3695829951343408861029541, 2655840753566898560252100 542, 139028384968680306034023200, 36879659147456455555864081 543, 130750714824264730429, 27089515171392307500 546, 7, 6 551, 122500, 32041 557, 1698956704241831560325, 1616556585387687403684 559, 2608225, 4489 561, 17, 16 565, 949765, 311364 566, 19462073861042, 258756359761 573, 72524591339947200, 49616483059370929 574, 288, 1 581, 306916, 269667 582, 194, 169 583, 4127673125, 4045752524 589, 16931584, 5130225 590, 16810, 4489 591, 1976836, 32761 597, 20332217085474456484, 12746213318246067075 598, 899496, 528529 599, 3038870911771011886354576369729571392520234010000, 1684988316181288974002607111763585731071026338361 602, 25, 18 606, 2187, 338 607, 125293941159013578901968618425335451919780530015678850455884723272256, 28106647023546616494565410735723361950369596834744611141477253594769 609, 25, 4 613, 134258586449534325996866004325, 2167741978776656970024461796 614, 7188810031138898, 3271718393929969 615, 484, 361 622, 16200, 961 623, 9246113759335228416, 917852707195412209 629, 629, 100 631, 7121391825113218180095235600, 7063072218419717533435319241 638, 44839971, 38281250 645, 125, 4 646, 18, 1 647, 1322160612567067320518390399934355553879100499050449996991864027974293776, 582675374825367622310345719733993297770816424257895824163140848520889449 651, 28, 3 653, 16039906351002089646726601004613956083758645547269701746398265925, 14182059891839277799515376046557567884269244616864333585635276676 654, 36842, 14641 655, 250986125, 4150604 658, 128, 47 661, 1827011937314914952625145627525, 1788754861111702406537568159684 662, 3964067407661153562050, 2971995056622029919649 663, 325, 204 669, 196, 27 670, 603, 242 671, 36, 25 674, 337, 288 677, 47927715455890725716869695510411087930312812511079044688038242680325, 37092798385418402372420837363147418226160953779371767824311918038756 678, 7106450, 6589489 679, 23324765703184, 22433058686025 685, 101380695056485, 14140593118404 687, 7502269, 410700 689, 689, 400 694, 144722, 108241 695, 57924313445, 26709551116 701, 424441571475461208113369493075504899525, 415387564510365861372968392912738762084 703, 1384862947204, 774007569729 709, 204901, 44100 710, 1440, 1369 717, 5402982087403200, 442947484849 718, 40949496158275253352971518245000, 6499888481560323372961451565601 719, 350438400, 272613121 721, 112, 9 723, 10609, 1200 727, 262866283445837565478197874923005550901524713509432228282087904339282031376, 134743807384934886048168596694816992039931007003491174422071019408679945849 731, 1521, 688 733, 42202047131581996388978204646062320777476945363051718065925, 30875705726293957067602327680955336278845752827135827134404 734, 829472, 18769 741, 16, 3 742, 1991056936, 675116289 743, 442081912085141580068205924381130314284533683543584687891856, 4923545999394502506171933125524998796854945340571916996169 749, 207152, 94249 751, 270400, 53361 757, 13602273107694756028798442021854633219790498492407291327120413272088837819189925, 6244756715574614879455468252566469219452570766747039517820858029581943762708004 758, 29448487134996834815262304148851250, 186062309862174127714929896033329 759, 12, 11 761, 640001, 1600 766, 691932401810510112, 23501280303420529 767, 211010428335204, 14971660478329 773, 989084230222373972333910805925, 395556919548784859614258538116 777, 148, 27 781, 4324112302500, 3683537773619 782, 24200, 5041 789, 8489443248, 6945388921 790, 77440, 1521 791, 113, 112 793, 19825, 17424 797, 33774422602919140585730411489969262195586409011795565639723428586290973904287392325, 5996643567874741587549707954567755873065370996868154128163410594841412962150844516 798, 32, 25 799, 64, 17 805, 81, 80 806, 31, 18 807, 415681879805284, 40681805176441 813, 196, 75 814, 74, 25 815, 4007741050385284, 3642335055801121 821, 672419525, 37429924 822, 63685236050, 49283556001 823, 3242635455264585956601518619150703534549412676347725149402642778377509709168016, 1195374870805855749961759118845148692286556133409866986910805318758576537878009 829, 1030055240843789503113650334926694761339101122026725, 177750573637318822104546934614413436162693927477444 830, 6723, 1682 831, 6650436108390733, 529824935498508 838, 128300399536533744621494450, 3013299452291575001275249 839, 921024090000, 798612109801 853, 2063644701214713560377801387332325, 1714573714638475303870575201815556 854, 11767, 1458 861, 2541, 2500 862, 2016643668909039569085152743098413287200, 1974330716960706704228129795667231395761 863, 1865955041348432071713419570491743304896576, 1858782575121402159177467199382221005546449 866, 361, 72 869, 475904, 286225 870, 145, 24 871, 13293316, 12809241 877, 8426023978582911503830444690267601350433831421925, 81307125890381364089453146670538087002747953764 878, 4892629929800, 4646443224721 879, 3844, 3481 885, 399424, 205379 886, 6568985816978, 4971575468209 887, 4536847899397194814782383112783728104152935409787863620887731098256, 376228287896231431732378232468716810769990224188408143567460976969 889, 64, 63 890, 49, 40 893, 98976113486833636, 3224135528266411 894, 1391283, 10658 895, 142805, 86636 901, 2873, 848 902, 5281250, 5276209 903, 172, 3 905, 181, 180 910, 65, 56 911, 10337603501168361479121072317589371823133263769269309465600, 10251038973842988792231450622875404412229672018176136927201 915, 61, 60 917, 6243557649986596, 1330622224234371 919, 172438957409290000, 94730381860226121 926, 86485853224649872668557955872, 4336954975876583369002712209 933, 1180457237875251598830000, 821142362599087465303081 934, 37538, 289 935, 2704, 2209 941, 45445208211554090721215702021, 45093936839814465880092660100 942, 1673306, 1190281 943, 34481, 9200 949, 949, 900 951, 196, 121 957, 29, 4 958, 8227224286063797798196820892172800, 7536459073489487535964603134454081 959, 5184, 4225 965, 1368547063155978485, 72225290780597956 966, 23, 2 967, 1669158917895536251704589403770563104478686864745616, 991936373451122630263885658159644238540608455699609 973, 5476, 1251 974, 2927411969212856264181512, 129458146762062069194929 982, 22599162783624879100060329302450, 1855038663733663467087769013329 983, 34499614166703093964327810353786154232194081202481758752971756816, 27595011125826459222677977624378565666052727191819391300400129209 985, 26569, 6724 987, 336, 289 989, 44548856356, 23518926067 991, 139113978859943328654400, 126099694516817139337521 995, 320, 121 997, 42213709768307514171686429890363488527317316427348844504307265329655015861152197726745537308248325, 3761555284456864241854415331157386556136120253716483379037244121900171126528942748431935644922916 998, 99017507481041765919078929839247234450, 45684092521200925325386025716112737489
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