The key to this problem is that for A2 + B2 = C2 we have that A2 = C2 - B2 = (C-B)(C+B)

So rather than searching upward from A when looking for possible B values, we look for integral values X = (C-B) and Y = (C+B) such that A2 = XY. We can afford to enumerate over all 1 ≤ X < A to locate such factor pairs.

For each such pair X < Y, if it is the case that X=C-B, Y=C+B, then B = (Y-X)/2 and C = (Y+X)/2 . We need only verify that (Y-X)/2 is integral and that it is strictly larger than A. As an optimization, to ensure that B > A, we can limit the search to 1 ≤ X < A(√2 - 1).

For far more in depth discussion of related mathematical properties, see the entry for Pythagorean triples in Wikipedia.

For those interested in solutions for the test data, the following are the (B,C) pairs for each A.

  1. A=3
    1 solution: (4, 5)

  2. A=12
    2 solutions: (16, 20), (35, 37)

  3. A=2
    0 solutions

  4. A=1048574
    1 solution: (274876858368, 274876858370)

  5. A=1048575
    175 solutions: (1051732, 1485143) (1241240, 1624865) (1264400, 1642625) (1291500, 1663575) (1345320, 1705695) (1373008, 1727617) (1398100, 1747625) (1401516, 1750359) (1779400, 2065375) (1809412, 2091287) (1813500, 2094825) (1843380, 2120745) (1948100, 2212375) (1979660, 2240215) (2268448, 2499073) (2309724, 2536599) (2350940, 2574185) (2434400, 2650625) (2472932, 2686057) (2511260, 2721385) (2516580, 2726295) (2601112, 2804513) (2692060, 2889065) (3065528, 3239903) (3166020, 3335145) (3172540, 3341335) (3220228, 3386647) (3380860, 3539735) (3387780, 3546345) (3438396, 3594729) (3497792, 3651583) (3970540, 4106665) (4235220, 4363095) (4297500, 4423575) (4306148, 4431977) (4370600, 4494625) (4516460, 4636585) (4591836, 4710039) (4669672, 4785953) (5366900, 5468375) (5719500, 5814825) (5730868, 5826007) (5815600, 5909375) (5899960, 5992415) (7112844, 7189719) (7126900, 7203625) (7229740, 7305385) (7231672, 7307297) (7350480, 7424895) (7591500, 7663575) (7829360, 7899265) (8853988, 8915863) (9429084, 9487209) (9583340, 9640535) (9723560, 9779935) (9742680, 9798945) (9882576, 9938049) (10374760, 10427615) (10523632, 10575743) (11895740, 11941865) (12093120, 12138495) (12290488, 12335137) (12690940, 12734185) (12876160, 12918785) (13060600, 13102625) (13086216, 13128159) (13933112, 13972513) (15746140, 15781015) (16236000, 16269825) (16267808, 16301567) (17285600, 17317375) (17319456, 17351169) (20179400, 20206625) (21441032, 21466657) (21483000, 21508575) (21790080, 21815295) (22150660, 22175465) (22870600, 22894625) (26644464, 26665089) (27078040, 27098335) (28364752, 28384127) (28826280, 28845345) (29303060, 29321815) (29721692, 29740183) (35748720, 35764095) (35818640, 35833985) (36330248, 36345377) (36339860, 36354985) (36930996, 36945879) (38130480, 38144895) (39314572, 39328553) (44418440, 44430815) (47284920, 47296545) (48053968, 48065407) (48753100, 48764375) (48848436, 48859689) (52000652, 52011223) (59589400, 59598625) (60574500, 60583575) (63558488, 63567137) (64483100, 64491625) (65403860, 65412265) (78814400, 78821375) (79960892, 79967767) (81261180, 81267945) (86504260, 86510615) (97731368, 97736993) (100962340, 100967785) (107266660, 107271785) (107476380, 107481495) (109010916, 109015959) (110812832, 110817793) (114410660, 114415465) (133271820, 133275945) (135438908, 135442967) (141870260, 141874135) (144177156, 144180969) (146560312, 146564063) (178780500, 178783575) (179130028, 179133097) (181735600, 181738625) (190686996, 190689879) (222121900, 222124375) (236452500, 236454825) (243792560, 243794815) (293201604, 293203479) (297969140, 297970985) (302894280, 302896095) (322435960, 322437665) (327039472, 327041153) (394088740, 394090135) (399820960, 399822335) (406322136, 406323489) (432536552, 432537823) (488670340, 488671465) (504824768, 504825857) (536345600, 536346625) (537394176, 537395199) (572064832, 572065793) (666369000, 666369825) (709360600, 709361375) (879607312, 879607937) (893909880, 893910495) (908685260, 908685865) (1110615440, 1110615935) (1182268080, 1182268545) (1218968212, 1218968663) (1466012520, 1466012895) (1489850128, 1489850497) (1514475756, 1514476119) (1612183892, 1612184233) (1970447048, 1970447327) (1999108100, 1999108375) (2443354400, 2443354625) (2681730460, 2681730665) (3331846980, 3331847145) (3546804860, 3546805015) (4398038060, 4398038185) (4469550876, 4469550999) (4543427752, 4543427873) (5553078388, 5553078487) (5911341516, 5911341609) (7330063500, 7330063575) (9995541160, 9995541215) (12216772540, 12216772585) (13408652792, 13408652833) (16659235296, 16659235329) (17734024672, 17734024703) (21990190600, 21990190625) (36650317680, 36650317695) (49977705932, 49977705943) (61083862808, 61083862817) (109950953060, 109950953065) (183251588436, 183251588439) (549754765312, 549754765313)

  6. A=4
    0 solutions

  7. A=60
    9 solutions: (63, 87) (80, 100) (91, 109) (144, 156) (175, 185) (221, 229) (297, 303) (448, 452) (899, 901)

  8. A=101
    1 solution: (5100, 5101)

  9. A=540
    25 solutions: (567, 783) (629, 829) (720, 900) (819, 981) (897, 1047) (1155, 1275) (1296, 1404) (1408, 1508) (1575, 1665) (1989, 2061) (2400, 2460) (2673, 2727) (2891, 2941) (3625, 3665) (4032, 4068) (4845, 4875) (6063, 6087) (7280, 7300) (8091, 8109) (12144, 12156) (14575, 14585) (18221, 18229) (24297, 24303) (36448, 36452) (72899, 72901)

  10. A=840
    49 solutions: (882, 1218) (1026, 1326) (1053, 1347) (1081, 1369) (1120, 1400) (1274, 1526) (1350, 1590) (1463, 1687) (1575, 1785) (1664, 1864) (1702, 1898) (1870, 2050) (2016, 2184) (2125, 2285) (2277, 2427) (2378, 2522) (2450, 2590) (2737, 2863) (2880, 3000) (3094, 3206) (3478, 3578) (3551, 3649) (3627, 3723) (3875, 3965) (4158, 4242) (4370, 4450) (4864, 4936) (5005, 5075) (5850, 5910) (6272, 6328) (7031, 7081) (7326, 7374) (8379, 8421) (8800, 8840) (9782, 9818) (11009, 11041) (11745, 11775) (12586, 12614) (14688, 14712) (17630, 17650) (19591, 19609) (22042, 22058) (25193, 25207) (29394, 29406) (35275, 35285) (44096, 44104) (58797, 58803) (88198, 88202) (176399, 176401)

  11. A=990
    17 solutions: (1320, 1650) (1680, 1950) (1904, 2146) (2376, 2574) (2944, 3106) (3192, 3342) (4400, 4510) (5400, 5490) (7392, 7458) (9048, 9102) (9776, 9826) (16320, 16350) (22264, 22286) (27216, 27234) (49000, 49010) (81672, 81678) (245024, 245026)

  12. A=1000008
    290 solutions: (1052256, 1451640) (1058531, 1456195) (1067040, 1462392) (1136994, 1514190) (1147706, 1522250) (1155840, 1528392) (1240206, 1593150) (1333344, 1666680) (1343085, 1674483) (1425450, 1741242) (1447131, 1759035) (1519495, 1819033) (1527360, 1825608) (1538031, 1834545) (1550094, 1844670) (1638560, 1919608) (1649781, 1929195) (1652145, 1931217) (1744794, 2011050) (1756550, 2021258) (1765290, 2028858) (1862656, 2114120) (1875015, 2125017) (1877619, 2127315) (1888990, 2137358) (1992706, 2229550) (2007369, 2242665) (2020480, 2254408) (2123469, 2347155) (2138906, 2361130) (2152710, 2373642) (2266719, 2477505) (2269755, 2480283) (2284256, 2493560) (2291975, 2500633) (2404050, 2603742) (2415310, 2614142) (2430594, 2628270) (2447881, 2644265) (2556885, 2745483) (2576320, 2763592) (2591094, 2777370) (2728131, 2905635) (2745120, 2921592) (2757755, 2933467) (2916690, 3083358) (3087744, 3245640) (3128190, 3284142) (3278406, 3427530) (3321120, 3468408) (3487906, 3628430) (3513594, 3653130) (3689280, 3822408) (3711631, 3843985) (3728256, 3860040) (3913910, 4039642) (3942510, 4067358) (3964256, 4088440) (4163045, 4281467) (4191210, 4308858) (4216406, 4333370) (4430419, 4541875) (4444480, 4555592) (4471119, 4581585) (4697406, 4802670) (4725490, 4830142) (4757265, 4861233) (4957869, 5057715) (4979744, 5079160) (5009445, 5108283) (5043050, 5141242) (5293094, 5386730) (5321895, 5415033) (5328531, 5421555) (5589390, 5678142) (5622594, 5710830) (5647294, 5735150) (5958381, 6041715) (5965785, 6049017) (6293910, 6372858) (6335835, 6414267) (6373344, 6451320) (6668655, 6743217) (6752706, 6826350) (7081205, 7151467) (7089969, 7160145) (7131840, 7201608) (7478406, 7544970) (7555350, 7621242) (7922119, 7984985) (7978656, 8041080) (8021650, 8083742) (8421065, 8480233) (8470656, 8529480) (8520535, 8579017) (8944430, 9000158) (8972294, 9027850) (9473760, 9526392) (9529469, 9581795) (9592506, 9644490) (10034050, 10083758) (10128235, 10177483) (10159744, 10208840) (10656415, 10703233) (10726830, 10773342) (11245344, 11289720) (11311365, 11355483) (11325281, 11369345) (11360480, 11404408) (11993994, 12035610) (12647031, 12686505) (12730494, 12769710) (12805170, 12844158) (13439865, 13477017) (13560645, 13597467) (14232570, 14267658) (14316006, 14350890) (14410419, 14445075) (15006735, 15040017) (15160119, 15193065) (16004130, 16035342) (16089869, 16120915) (16109625, 16140633) (16886506, 16916090) (16985430, 17014842) (17930656, 17958520) (17986255, 18014033) (18008331, 18036075) (18986994, 19013310) (19112081, 19138225) (19224000, 19249992) (20105381, 20130235) (20293406, 20318030) (20356310, 20380858) (21363475, 21386867) (21488544, 21511800) (22523970, 22546158) (22683610, 22705642) (22753906, 22775870) (24019200, 24040008) (25490400, 25510008) (25639581, 25659075) (26907594, 26926170) (28491456, 28509000) (28658175, 28675617) (28846830, 28864158) (30450369, 30466785) (32031669, 32047275) (32242506, 32258010) (33795200, 33809992) (33992919, 34007625) (34034595, 34049283) (35882210, 35896142) (36037470, 36051342) (37993725, 38006883) (38243770, 38256842) (38467494, 38480490) (40369131, 40381515) (40605280, 40617592) (40731031, 40743305) (42744494, 42756190) (42994530, 43006158) (43277465, 43289017) (45064581, 45075675) (45383744, 45394760) (45524285, 45535267) (48054006, 48064410) (48370219, 48380555) (50995506, 51005310) (53829120, 53838408) (54061985, 54071233) (56996070, 57004842) (57706656, 57715320) (60913050, 60921258) (64496640, 64504392) (67601494, 67608890) (68080206, 68087550) (71774869, 71781835) (72085344, 72092280) (76497344, 76503880) (76944735, 76951233) (80747550, 80753742) (81219794, 81225950) (85497760, 85503608) (85997781, 86003595) (86563594, 86569370) (90775750, 90781258) (91372995, 91378467) (96115815, 96121017) (96748190, 96753358) (101998365, 102003267) (102123369, 102128265) (107665206, 107669850) (108130906, 108135530) (113998719, 114003105) (115419810, 115424142) (121123905, 121128033) (121832256, 121836360) (128999094, 129002970) (135208535, 135212233) (136165920, 136169592) (144175890, 144179358) (152999590, 153002858) (161499744, 161502840) (162444205, 162447283) (170999906, 171002830) (173131520, 173134408) (181555631, 181558385) (182750094, 182752830) (192903025, 192905617) (193500256, 193502840) (204250410, 204252858) (215333895, 215336217) (216265280, 216267592) (230842869, 230845035) (242250906, 242252970) (243667590, 243669642) (258001095, 258003033) (272334594, 272336430) (274126281, 274128105) (288354381, 288356115) (306001631, 306003265) (306376635, 306378267) (323001810, 323003358) (342002005, 342003467) (346265206, 346266650) (363377219, 363378595) (365502240, 365503608) (385807994, 385809290) (387002450, 387003742) (408502656, 408503880) (432532294, 432533450) (484503360, 484504392) (487336719, 487337745) (544670565, 544671483) (548253930, 548254842) (578712531, 578713395) (612754494, 612755310) (646004781, 646005555) (692531495, 692532217) (726755470, 726756158) (731005506, 731006190) (771616960, 771617608) (774005869, 774006515) (817006230, 817006842) (822381275, 822381883) (865065455, 865066033) (919132081, 919132625) (969007494, 969008010) (1096508544, 1096509000) (1157425710, 1157426142) (1225509600, 1225510008) (1453511456, 1453511800) (1462011525, 1462011867) (1543234406, 1543234730) (1634012919, 1634013225) (1644763006, 1644763310) (1736138745, 1736139033) (1838264570, 1838264842) (1938015375, 1938015633) (2193017430, 2193017658) (2314851744, 2314851960) (2451019506, 2451019710) (2907023170, 2907023342) (3086469055, 3086469217) (3289526240, 3289526392) (3472277706, 3472277850) (3676529344, 3676529480) (4386035031, 4386035145) (4629703650, 4629703758) (4902039165, 4902039267) (5208416619, 5208416715) (5814046469, 5814046555) (6579052594, 6579052670) (6944555520, 6944555592) (7353058790, 7353058858) (9259407381, 9259407435) (10416833310, 10416833358) (13158105245, 13158105283) (13889111094, 13889111130) (14706117631, 14706117665) (15625249985, 15625250017) (20833666656, 20833666680) (27778222215, 27778222233) (31250499994, 31250500010) (41667333330, 41667333342) (62501000000, 62501000008) (83334666669, 83334666675) (125002000006, 125002000010) (250004000015, 250004000017)

  13. A=1000692
    297 solutions: (1018381, 1427755) (1092025, 1481183) (1103235, 1489467) (1111481, 1495585) (1158145, 1530583) (1166619, 1537005) (1173744, 1542420) (1258085, 1607533) (1258895, 1608167) (1334256, 1667820) (1403769, 1723935) (1413315, 1731717) (1426425, 1742433) (1508144, 1809940) (1509056, 1810700) (1583631, 1873305) (1594005, 1882083) (1608255, 1894167) (1687280, 1961708) (1698144, 1971060) (1779369, 2041455) (1790675, 2051317) (1892400, 2140692) (1904256, 2151180) (1914231, 2160015) (1994069, 2231075) (2023405, 2257333) (2109665, 2334967) (2140369, 2362745) (2227731, 2442165) (2239056, 2452500) (2252640, 2464908) (2389269, 2590365) (2496144, 2689260) (2511040, 2703092) (2623995, 2808333) (2645265, 2828217) (2660931, 2842875) (2778256, 2952980) (2779744, 2954380) (2918685, 3085467) (2942055, 3107583) (2959269, 3123885) (3089856, 3247860) (3242635, 3393533) (3244345, 3395167) (3430944, 3573900) (3447519, 3589815) (3467405, 3608917) (3580381, 3717595) (3600975, 3737433) (3629285, 3764717) (3824681, 3953425) (3971019, 4095165) (3993705, 4117167) (4012800, 4135692) (4222256, 4339220) (4242381, 4358805) (4447520, 4558708) (4638195, 4744917) (4664481, 4770615) (4700619, 4805955) (4929360, 5029908) (5137125, 5233683) (5166119, 5262145) (5205981, 5301285) (5427744, 5519220) (5458320, 5549292) (5687555, 5774917) (5690465, 5777783) (5766481, 5852665) (6008256, 6091020) (6042000, 6124308) (6070405, 6152333) (6298215, 6377217) (6346525, 6424933) (6382119, 6460095) (6969105, 7040583) (7001760, 7072908) (7040944, 7111700) (7360144, 7427860) (7435935, 7502967) (7745920, 7810292) (8117769, 8179215) (8179731, 8240715) (8225385, 8286033) (8532235, 8590717) (8572080, 8630292) (8978431, 9034025) (9046869, 9102045) (9097295, 9152167) (9480240, 9532908) (9934131, 9984405) (10009769, 10059665) (10483440, 10531092) (10532285, 10579717) (10924095, 10969833) (10984869, 11030355) (11068431, 11113575) (11597600, 11640692) (12078585, 12119967) (12145731, 12186885) (12202256, 12243220) (12751856, 12791060) (12822720, 12861708) (12882375, 12921183) (14056881, 14092455) (14134955, 14170333) (14242305, 14277417) (14771075, 14804933) (14922144, 14955660) (15540119, 15572305) (15745015, 15776783) (15752965, 15784717) (16405200, 16435692) (16496256, 16526580) (17187819, 17216925) (17414355, 17443083) (18135120, 18162708) (18235744, 18263180) (18999981, 19026315) (19144119, 19170255) (19250325, 19276317) (20056960, 20081908) (21002619, 21026445) (21100144, 21123860) (22170720, 22193292) (22397181, 22419525) (23227519, 23249065) (24435235, 24455717) (24620505, 24640833) (24757019, 24777235) (25533115, 25552717) (25674681, 25694175) (25793856, 25813260) (27214175, 27232567) (28510944, 28528500) (29869425, 29886183) (30095835, 30112467) (31513856, 31529740) (31529744, 31545620) (32833269, 32848515) (33015255, 33030417) (33265485, 33280533) (34850256, 34864620) (36290931, 36304725) (36492065, 36505783) (38307840, 38320908) (38520144, 38533140) (38698869, 38711805) (40132631, 40145105) (40660855, 40673167) (42218075, 42229933) (42773731, 42785435) (44358369, 44369655) (44811120, 44822292) (47301231, 47311815) (49256256, 49266420) (49529200, 49539308) (51602265, 51611967) (52281369, 52290945) (54442144, 54451340) (57035055, 57043833) (57467205, 57475917) (57785631, 57794295) (60204144, 60212460) (63039625, 63047567) (63071395, 63079333) (66542256, 66549780) (67221335, 67228783) (69711285, 69718467) (70239455, 70246583) (73888619, 73895395) (76625481, 76632015) (77050035, 77056533) (77407440, 77413908) (81330944, 81337100) (85556240, 85562092) (89630619, 89636205) (90309681, 90315225) (94610400, 94615692) (98520135, 98525217) (99065981, 99071035) (99816519, 99821535) (104569920, 104574708) (108891185, 108895783) (110379619, 110384155) (114940944, 114945300) (115577760, 115582092) (116113855, 116118167) (120414525, 120418683) (121998981, 122003085) (133090155, 133093917) (134448256, 134451980) (140484256, 140487820) (141917805, 141921333) (147782320, 147785708) (154819731, 154822965) (156856875, 156860067) (162666505, 162669583) (171116869, 171119795) (172413231, 172416135) (173368445, 173371333) (180623520, 180626292) (189224769, 189227415) (199636800, 199639308) (209143431, 209145825) (210727869, 210730245) (220762640, 220764908) (229885155, 229887333) (231158769, 231160935) (232230944, 232233100) (244001040, 244003092) (268899305, 268901167) (270936435, 270938283) (280971185, 280972967) (283838256, 283840020) (295567181, 295568875) (299456245, 299457917) (313716144, 313717740) (331144905, 331146417) (344828640, 344830092) (346739056, 346740500) (361249119, 361250505) (366002415, 366003783) (399275481, 399276735) (421457520, 421458708) (425758119, 425759295) (441526981, 441528115) (464463505, 464464583) (470574881, 470575945) (488003619, 488004645) (517243565, 517244533) (541874256, 541875180) (567677835, 567678717) (598913744, 598914580) (627433485, 627434283) (632186775, 632187567) (662290944, 662291700) (689658369, 689659095) (693479195, 693479917) (732005856, 732006540) (772672885, 772673533) (812811769, 812812385) (842915931, 842916525) (851517120, 851517708) (941150560, 941151092) (993436731, 993437235) (1034487856, 1034488340) (1083749205, 1083749667) (1098009069, 1098009525) (1197828115, 1197828533) (1264374144, 1264374540) (1277275925, 1277276317) (1324582455, 1324582833) (1464012225, 1464012567) (1545346256, 1545346580) (1625624000, 1625624308) (1703034681, 1703034975) (1882301519, 1882301785) (1896561381, 1896561645) (1986873840, 1986874092) (2068976075, 2068976317) (2196018480, 2196018708) (2318019519, 2318019735) (2528748585, 2528748783) (2554552144, 2554552340) (2980310865, 2980311033) (3090692755, 3090692917) (3251248231, 3251248385) (3294027815, 3294027967) (3793122960, 3793123092) (3973747869, 3973747995) (4392037131, 4392037245) (4636039200, 4636039308) (5109104435, 5109104533) (5689684495, 5689684583) (5960621856, 5960621940) (6588055744, 6588055820) (6954058845, 6954058917) (7586246019, 7586246085) (8940932819, 8940932875) (9272078481, 9272078535) (11379369056, 11379369100) (11921243775, 11921243817) (13176111545, 13176111583) (13908117744, 13908117780) (17881865680, 17881865708) (20862176631, 20862176655) (22758738145, 22758738167) (27816235515, 27816235533) (35763731381, 35763731395) (41724353280, 41724353292) (62586529925, 62586529933) (83448706569, 83448706575) (125173059856, 125173059860) (250346119715, 250346119717)

  14. A=1000800
    298 solutions: (1034044, 1439044) (1050840, 1451160) (1102790, 1489210) (1146750, 1522050) (1209856, 1570144) (1211112, 1571112) (1229455, 1585295) (1266111, 1613889) (1267402, 1614902) (1276275, 1621875) (1334400, 1668000) (1383680, 1707680) (1403622, 1723878) (1405001, 1725001) (1444905, 1757655) (1465432, 1774568) (1517880, 1818120) (1594890, 1882890) (1662440, 1940440) (1719816, 1989816) (1743060, 2009940) (1802500, 2061700) (1876500, 2126700) (1964904, 2205096) (1966668, 2206668) (2044074, 2275926) (2111410, 2336590) (2140600, 2363000) (2210520, 2426520) (2238873, 2452377) (2297670, 2506170) (2311445, 2518805) (2371838, 2574338) (2401920, 2602080) (2495395, 2688605) (2512335, 2704335) (2574975, 2762625) (2689928, 2870072) (2692224, 2872224) (2725790, 2903710) (2795429, 2969179) (2811750, 2984550) (2919000, 3085800) (3010360, 3172360) (3047436, 3207564) (3050002, 3210002) (3162716, 3317284) (3260940, 3411060) (3405780, 3549780) (3533380, 3672380) (3642132, 3777132) (3686280, 3819720) (3799400, 3929000) (3940650, 4065750) (4109952, 4230048) (4113336, 4233336) (4262037, 4377963) (4289625, 4404825) (4391705, 4504295) (4448000, 4559200) (4583040, 4691040) (4637874, 4744626) (4751715, 4855965) (4778410, 4882090) (4895551, 4996801) (4953960, 5054040) (5168670, 5264670) (5514964, 5605036) (5519448, 5609448) (5585020, 5673980) (5753100, 5839500) (5963100, 6046500) (6142220, 6223220) (6214968, 6295032) (6220004, 6300004) (6441358, 6518642) (6634470, 6709530) (6919560, 6991560) (7001291, 7072459) (7171010, 7240510) (7210815, 7279935) (7385514, 7453014) (7472640, 7539360) (7696000, 7760800) (7793005, 7857005) (7975125, 8037675) (8309976, 8370024) (8316672, 8376672) (8665650, 8723250) (8979400, 9035000) (9247080, 9301080) (9355812, 9409188) (9634580, 9686420) (9982980, 10033020) (10409340, 10457340) (11097482, 11142518) (11106396, 11151396) (11236760, 11281240) (11571000, 11614200) (11988750, 12030450) (12054889, 12096361) (12345190, 12385690) (12489984, 12530016) (12500008, 12540008) (12940679, 12979321) (13022475, 13060875) (13325235, 13362765) (13893120, 13929120) (14055958, 14091542) (14394145, 14428895) (14473470, 14508030) (14821653, 14855403) (14995320, 15028680) (15440600, 15473000) (15634010, 15666010) (16664988, 16695012) (16678344, 16708344) (17374500, 17403300) (18000500, 18028300) (18534660, 18561660) (18751656, 18778344) (19308040, 19333960) (20003490, 20028510) (20854680, 20878680) (21724605, 21747645) (22228741, 22251259) (22246542, 22269042) (22506880, 22529120) (23174400, 23196000) (24008775, 24029625) (24140882, 24161618) (24720755, 24741005) (25009992, 25030008) (25030016, 25050016) (26073750, 26092950) (27813240, 27831240) (28138604, 28156396) (28972860, 28990140) (30015660, 30032340) (30905500, 30921700) (31292020, 31308020) (33352494, 33367506) (33379188, 33394188) (34770600, 34785000) (36021850, 36035750) (36219963, 36233787) (37089570, 37103070) (37523328, 37536672) (38635520, 38648480) (39118625, 39131425) (40025745, 40038255) (41727360, 41739360) (43466490, 43478010) (44509959, 44521209) (45030440, 45041560) (46365000, 46375800) (48297316, 48307684) (50034996, 50045004) (50075032, 50085032) (52161900, 52171500) (55639980, 55648980) (56290552, 56299448) (57958680, 57967320) (60043830, 60052170) (61823150, 61831250) (62596040, 62604040) (65204535, 65212215) (66716247, 66723753) (66769626, 66777126) (69552000, 69559200) (72054125, 72061075) (72450294, 72457206) (74189265, 74196015) (75056664, 75063336) (77280760, 77287240) (78246850, 78253250) (83463720, 83469720) (86941620, 86947380) (90069220, 90074780) (92738100, 92743500) (96602408, 96607592) (100077498, 100082502) (100157564, 100162564) (104331000, 104335800) (108678321, 108682929) (111286710, 111291210) (112587776, 112592224) (115923840, 115928160) (120093915, 120098085) (123652375, 123656425) (125198080, 125202080) (130414830, 130418670) (133544877, 133548627) (139109400, 139113000) (144905772, 144909228) (150118332, 150121668) (154566380, 154569620) (156498500, 156501700) (166931940, 166934940) (173887560, 173890440) (180142610, 180145390) (185480250, 185482950) (193208704, 193211296) (195623845, 195626405) (200158749, 200161251) (200318878, 200321378) (208665600, 208668000) (217360098, 217362402) (222576795, 222579045) (225178888, 225181112) (231850920, 231853080) (250399160, 250401160) (260832540, 260834460) (278221500, 278223300) (289814136, 289815864) (300239166, 300240834) (309135190, 309136810) (312999400, 313001000) (326041107, 326042643) (333866130, 333867630) (347777280, 347778720) (360287305, 360288695) (370962525, 370963875) (386419352, 386420648) (391249610, 391250890) (400639631, 400640881) (417333000, 417334200) (434721924, 434723076) (450359444, 450360556) (463703460, 463704540) (500799820, 500800820) (521666520, 521667480) (556444350, 556445250) (579629568, 579630432) (600479583, 600480417) (618271595, 618272405) (626000000, 626000800) (652083366, 652084134) (667733385, 667734135) (695555640, 695556360) (772839676, 772840324) (782500180, 782500820) (834666900, 834667500) (869444712, 869445288) (900719722, 900720278) (927407730, 927408270) (978125369, 978125881) (1001600390, 1001600890) (1043333760, 1043334240) (1112889375, 1112889825) (1159259784, 1159260216) (1252000600, 1252001000) (1304167308, 1304167692) (1391111820, 1391112180) (1545679838, 1545680162) (1565000840, 1565001160) (1669334250, 1669334550) (1738889856, 1738890144) (1801439861, 1801440139) (1854815865, 1854816135) (1956251122, 1956251378) (2003201155, 2003201405) (2086667880, 2086668120) (2318519892, 2318520108) (2504001500, 2504001700) (2608334904, 2608335096) (2782223910, 2782224090) (3091359919, 3091360081) (3130001920, 3130002080) (3338668725, 3338668875) (3477779928, 3477780072) (3912502436, 3912502564) (4173335940, 4173336060) (4637039946, 4637040054) (5008003150, 5008003250) (5216669952, 5216670048) (5564447955, 5564448045) (6260003960, 6260004040) (6955559964, 6955560036) (7825004968, 7825005032) (8346671970, 8346672030) (9274079973, 9274080027) (10016006375, 10016006425) (10433339976, 10433340024) (12520007980, 12520008020) (13911119982, 13911120018) (15650009984, 15650010016) (16693343985, 16693344015) (20866679988, 20866680012) (25040015990, 25040016010) (27822239991, 27822240009) (31300019992, 31300020008) (41733359994, 41733360006) (50080031995, 50080032005) (62600039996, 62600040004) (83466719997, 83466720003) (125200079998, 125200080002) (250400159999, 250400160001)

  15. A=1000960
    199 solutions: (1018248, 1427848) (1051008, 1451392) (1140984, 1517816) (1169280, 1539200) (1212780, 1572500) (1264800, 1612960) (1310400, 1648960) (1341153, 1673503) (1364970, 1692650) (1445136, 1757936) (1485519, 1791281) (1544832, 1840768) (1554432, 1848832) (1596912, 1884688) (1659336, 1937864) (1714176, 1985024) (1751220, 2017100) (1876800, 2127040) (2009280, 2244800) (2051184, 2282384) (2193408, 2411008) (2248848, 2461552) (2261688, 2473288) (2343696, 2548496) (2402304, 2602496) (2564592, 2753008) (2616000, 2800960) (2695350, 2875210) (2790720, 2964800) (2874720, 3044000) (2975700, 3139540) (3124872, 3281272) (3311616, 3459584) (3329664, 3476864) (3409656, 3553544) (3527568, 3666832) (3631488, 3766912) (3701850, 3834790) (3756489, 3887561) (3941280, 4066400) (4195200, 4312960) (4275768, 4391368) (4550016, 4658816) (4657224, 4763576) (4682076, 4787876) (4840992, 4943392) (4954752, 5054848) (5270496, 5364704) (5370720, 5463200) (5525595, 5615525) (5712000, 5799040) (5876400, 5961040) (6074280, 6156200) (6367044, 6445244) (6734208, 6808192) (6769728, 6843328) (6927228, 6999172) (7159584, 7229216) (7364544, 7432256) (7503405, 7569875) (7611282, 7676818) (7976400, 8038960) (8478720, 8537600) (8638236, 8696036) (9181632, 9236032) (9394212, 9447388) (9443502, 9496402) (9758784, 9809984) (9984576, 10034624) (10611648, 10658752) (10810800, 10857040) (11489280, 11532800) (11816280, 11858600) (12210000, 12250960) (12792738, 12831838) (13523904, 13560896) (13594656, 13631456) (13908414, 13944386) (14371392, 14406208) (14779872, 14813728) (15271716, 15304484) (15999720, 16031000) (17001600, 17031040) (17319822, 17348722) (18404064, 18431264) (18828306, 18854894) (18926679, 18953129) (19555968, 19581568) (20006688, 20031712) (21258624, 21282176) (21656280, 21679400) (23011200, 23032960) (23664300, 23685460) (24450720, 24471200) (25614801, 25634351) (27075552, 27094048) (27216912, 27235312) (27843807, 27861793) (28768896, 28786304) (29585136, 29602064) (30568008, 30584392) (32022900, 32038540) (34025280, 34040000) (34661319, 34675769) (36828528, 36842128) (37676553, 37689847) (39131136, 39143936) (40032144, 40044656) (42534912, 42546688) (43329900, 43341460) (46038720, 46049600) (47344470, 47355050) (48916800, 48927040) (54164976, 54174224) (54447624, 54456824) (57550848, 57559552) (59182968, 59191432) (61148304, 61156496) (64057530, 64065350) (68061600, 68068960) (73667256, 73674056) (78271872, 78278272) (80073672, 80079928) (85078656, 85084544) (86668470, 86674250) (92085600, 92091040) (94696875, 94702165) (97841280, 97846400) (108336888, 108341512) (108902148, 108906748) (115108224, 115112576) (118372284, 118376516) (122302752, 122306848) (128120925, 128124835) (136128720, 136132400) (147339612, 147343012) (156548544, 156551744) (160152036, 160155164) (170161728, 170164672) (173341275, 173344165) (184175280, 184178000) (195686400, 195688960) (216677244, 216679556) (217807746, 217810046) (230219712, 230221888) (236747742, 236749858) (244608576, 244610624) (272260200, 272262040) (294681774, 294683474) (313099488, 313101088) (320306418, 320307982) (340325664, 340327136) (368352600, 368353960) (391374720, 391376000) (433356222, 433357378) (435617217, 435618367) (460441056, 460442144) (473497071, 473498129) (489218688, 489219712) (544521780, 544522700) (589364823, 589365673) (626200176, 626200976) (640614009, 640614791) (680652432, 680653168) (736706220, 736706900) (782750400, 782751040) (866713311, 866713889) (920882928, 920883472) (978438144, 978438656) (1089044250, 1089044710) (1252400952, 1252401352) (1361305416, 1361305784) (1473412950, 1473413290) (1565501280, 1565501600) (1841766264, 1841766536) (1956876672, 1956876928) (2178088845, 2178089075) (2504802204, 2504802404) (2722611108, 2722611292) (2946826155, 2946826325) (3131002800, 3131002960) (3683532732, 3683532868) (3913753536, 3913753664) (5009604558, 5009604658) (5445222354, 5445222446) (6262005720, 6262005800) (7367065566, 7367065634) (7827507168, 7827507232) (10019209191, 10019209241) (10890444777, 10890444823) (12524011500, 12524011540) (14734131183, 14734131217) (15655014384, 15655014416) (25048023030, 25048023050) (31310028792, 31310028808) (50096046075, 50096046085) (62620057596, 62620057604) (125240115198, 125240115202) (250480230399, 250480230401)

  16. A=1047658
    1 solution: (274396821240, 274396821242)

  17. A=1047659
    4 solutions: (8874288, 8935915) (1898943420, 1898943709) (32282040588, 32282040605) (548794690140, 548794690141)

  18. A=1048320
    863 solutions: (1053760, 1486400) (1062600, 1492680) (1076933, 1502917) (1077300, 1503180) (1086240, 1509600) (1093976, 1515176) (1100736, 1520064) (1117600, 1532320) (1120079, 1534129) (1124812, 1537588) (1136722, 1546322) (1144000, 1551680) (1151952, 1557552) (1161216, 1564416) (1168196, 1569604) (1176240, 1575600) (1183656, 1581144) (1185998, 1582898) (1201200, 1594320) (1216512, 1605888) (1226064, 1613136) (1241954, 1625246) (1246596, 1628796) (1262240, 1640800) (1272000, 1648320) (1280448, 1654848) (1287832, 1660568) (1306260, 1674900) (1306669, 1675219) (1314144, 1681056) (1322760, 1687800) (1332800, 1695680) (1349088, 1708512) (1353968, 1712368) (1370838, 1725738) (1381104, 1733904) (1397760, 1747200) (1414544, 1760656) (1417152, 1762752) (1425018, 1769082) (1425450, 1769430) (1442448, 1783152) (1453056, 1791744) (1462240, 1799200) (1484704, 1817504) (1490312, 1822088) (1493260, 1824500) (1513512, 1841112) (1521728, 1847872) (1531200, 1855680) (1542240, 1864800) (1560156, 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  19. A=1048532
    3 solutions: (68713709685, 68713709693) (137427419376, 137427419380) (274854838755, 274854838757)

  20. A=1048560
    253 solutions: (1100988, 1520412) (1112900, 1529060) (1131314, 1542514) (1189053, 1585347) (1201475, 1594685) (1300420, 1670500) (1398080, 1747600) (1411776, 1758576) (1484761, 1817689) (1590316, 1904884) (1628352, 1936752) (1701854, 1998946) (1708793, 2004857) (1826242, 2105858) (1842750, 2120190) (1948702, 2212898) (1966050, 2228190) (1983518, 2243618) (2104830, 2351550) (2123266, 2368066) (2261086, 2492386) (2262164, 2493364) (2407319, 2625769) (2516544, 2726256) (2537920, 2746000) (2571028, 2776628) (2878400, 3063440) (2879725, 3064685) (3058300, 3233060) (3083652, 3257052) (3219218, 3385682) (3260045, 3424525) (3286899, 3450099) (3416558, 3573842) (3488004, 3642204) (3626527, 3775073) (3639634, 3787666) (3862196, 4002004) (3893580, 4032300) (4095551, 4227649) (4128705, 4259775) (4162111, 4292161) (4394700, 4518060) (4430132, 4552532) (4695647, 4811297) (4697728, 4813328) (4898187, 5009163) (5190372, 5295228) (5231900, 5335940) (5296256, 5399056) (5521131, 5619819) (5565205, 5663125) (5895580, 5988100) (5898170, 5990650) (6247670, 6335050) (6297354, 6384054) (6563284, 6646516) (6643450, 6725690) (6696198, 6777798) (6951079, 7029721) (7091658, 7168758) (7390292, 7464308) (7829248, 7899152) (7891200, 7960560) (8881920, 8943600) (8952064, 9013264) (9482156, 9539956) (9879606, 9935094) (10078297, 10132697) (10459386, 10511814) (10541830, 10593850) (10669612, 10721012) (11116278, 11165622) (11203850, 11252810) (11860550, 11906810) (11865700, 11911940) (12560875, 12604565) (12659733, 12703083) (13188992, 13230608) (13348580, 13389700) (13453596, 13494396) (14241141, 14279691) (14836096, 14873104) (14842529, 14879521) (15710924, 15745876) (15834420, 15869100) (16694977, 16727873) (16826175, 16858815) (17810100, 17840940) (17950028, 17980628) (19007662, 19036562) (19073761, 19102561) (19800828, 19828572) (20197394, 20224594) (20958093, 20984307) (21122675, 21148685) (21377774, 21403474) (22269564, 22294236) (22444420, 22468900) (23755795, 23778925) (23766080, 23789200) (26409196, 26430004) (26728000, 26748560) (26937792, 26958192) (28061033, 28080617) (29699948, 29718452) (29712802, 29731298) (31448062, 31465538) (31694850, 31712190) (33414626, 33431074) (33676830, 33693150) (35643330, 35658750) (35923006, 35938306) (38036999, 38051449) (38169122, 38183522) (39622464, 39636336) (40415188, 40428788) (42774823, 42787673) (44557632, 44569968) (44907200, 44919440) (47549500, 47561060) (50522045, 50532925) (52833998, 52844402) (53471420, 53481700) (53890884, 53901084) (56136754, 56146546) (57259683, 57269283) (59413774, 59423026) (59439476, 59448724) (62909231, 62917969) (63402705, 63411375) (66841588, 66849812) (67365900, 67374060) (71298225, 71305935) (71857487, 71865137) (76349044, 76356244) (79255332, 79262268) (80840576, 80847376) (84209211, 84215739) (89124516, 89130684) (89823580, 89829700) (95107670, 95113450) (95437925, 95443685) (101052250, 101057690) (105675799, 105681001) (106950550, 106955690) (107789418, 107794518) (112280852, 112285748) (114526566, 114531366) (118834487, 118839113) (118885888, 118890512) (133689344, 133693456) (134737920, 134742000) (152703488, 152707088) (158515866, 158519334) (161686252, 161689652) (168423318, 168426582) (171791849, 171795049) (178253658, 178256742) (179651750, 179654810) (190219675, 190222565) (190880170, 190883050) (202108580, 202111300) (213904955, 213907525) (215582661, 215585211) (224565376, 224567824) (229056732, 229059132) (237775244, 237777556) (252636337, 252638513) (267381772, 267383828) (269478900, 269480940) (286321455, 286323375) (305409676, 305411476) (317034333, 317036067) (323375054, 323376754) (336849084, 336850716) (343586098, 343587698) (356509629, 356511171) (359305795, 359307325) (381762500, 381763940) (404219200, 404220560) (449132588, 449133812) (458115264, 458116464) (475552222, 475553378) (477203449, 477204601) (505274306, 505275394) (534765086, 534766114) (538959330, 538960350) (572644350, 572645310) (610820702, 610821602) (646751383, 646752233) (673699392, 673700208) (687173396, 687174196) (763526080, 763526800) (808439420, 808440100) (858966925, 858967565) (898266094, 898266706) (916231428, 916232028) (951105311, 951105889) (954407762, 954408338) (1010549428, 1010549972) (1069530943, 1069531457) (1077919425, 1077919935) (1145289420, 1145289900) (1221642079, 1221642529) (1347399396, 1347399804) (1374347392, 1374347792) (1431611883, 1431612267) (1527052700, 1527053060) (1616879350, 1616879690) (1717934330, 1717934650) (1796532647, 1796532953) (1832463306, 1832463606) (1908815956, 1908816244) (2021099264, 2021099536) (2290579200, 2290579440) (2694799098, 2694799302) (2748695084, 2748695284) (2863224054, 2863224246) (3054105670, 3054105850) (3233758955, 3233759125) (3435868900, 3435869060) (3664926837, 3664926987) (3817632128, 3817632272) (4042198732, 4042198868) (4294836161, 4294836289) (4581158580, 4581158700) (5389598349, 5389598451) (5497390318, 5497390418) (5726448252, 5726448348) (6108211475, 6108211565) (6871737920, 6871738000) (7635264364, 7635264436) (8084397566, 8084397634) (8589672418, 8589672482) (9162317250, 9162317310) (10994780711, 10994780761) (11452896576, 11452896624) (13743475900, 13743475940) (15270528782, 15270528818) (16168795183, 16168795217) (17179344884, 17179344916) (18324634545, 18324634575) (22905793188, 22905793212) (27486951830, 27486951850) (30541057591, 30541057609) (34358689792, 34358689808) (45811586394, 45811586406) (54973903675, 54973903685) (68717379596, 68717379604) (91623172797, 91623172803) (137434759198, 137434759202) (274869518399, 274869518401)