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Integral Points on Conic Curves: x^2-Dy^2=±1(D \in [1000,1999])


[2003.08.16]x2-Dy2=±1 (D \in [1000,1999])の整点


1000≦D≦1999かつ平方数でない整数Dについて、Pell方程式
     x2-Dy2 = ±1
の基本解を求めと、以下のようになる。


D fundamental solution [x,y] of x2-Dy2=-1 fundamental solution [x,y] of x2-Dy2=1
1000 - [39480499, 1248483]
1001 - [1060905, 33532]
1002 - [206869247, 6535248]
1003 - [9026, 285]
1004 - [27009633024199, 852416459730]
1005 - [2950149761, 93059568]
1006 - [14125267563780214605455, 445346140255574921748]
1007 - [476, 15]
1008 - [127, 4]
1009 [540, 17] [583201, 18360]
1010 [1303, 41] [3395619, 106846]
1011 - [8426, 265]
1012 - [3222617399, 101302110]
1013 [393166618, 12352985] [309159979019115849, 9713562669309460]
1014 - [4656965, 146246]
1015 - [352871, 11076]
1016 - [255, 8]
1017 - [2900932297217, 90965584992]
1018 [87050499, 2728333] [15155578752298003, 475005498176334]
1019 - [6089923321730, 190776436539]
1020 - [511, 16]
1021 [315217280372584882515030, 9865001296666956406909] [198723867690977573219668252231077415636351801801, 6219237759214762827187409503019432615976684540]
1022 - [1023, 32]
1023 - [32, 1]
1025 [32, 1] [2049, 64]
1026 - [1025, 32]
1027 - [133150393, 4154868]
1028 - [513, 16]
1029 - [97379496466615, 3035706895884]
1030 - [884307011291, 27553987434]
1031 - [651737448664200, 20297537082877]
1032 - [257, 8]
1033 [186860367596174196, 5813893046024093] [69833593956354694952807071952492833, 2172772383489805842331071281808456]
1034 - [15885, 494]
1035 - [1126, 35]
1036 - [847225, 26322]
1037 [2086882, 64805] [8710152963849, 270480776020]
1038 - [15404267, 478126]
1039 - [493194634979076339348637520, 15300674275156671864306921]
1040 - [129, 4]
1041 - [770918649927575, 23893687943492]
1042 [807, 25] [1302499, 40350]
1043 - [576927, 17864]
1044 - [192119201, 5945940]
1045 - [456191, 14112]
1046 - [3987042510565, 123277842162]
1047 - [136807, 4228]
1048 - [104980517, 3242859]
1049 [1536350825620, 47435409493] [4720747718766511296768801, 145754861076386671221320]
1050 - [8749, 270]
1051 - [955382766460015065912973250, 29469722410666558657086507]
1052 - [38713200767, 1193579112]
1053 - [649, 20]
1054 - [3355576256056895, 103358645005416]
1055 - [1689, 52]
1056 - [65, 2]
1057 - [239637339568529407, 7370840200132992]
1058 - [131836323, 4053146]
1059 - [5983486610, 183868081]
1060 - [73738369, 2264856]
1061 [9241228697111530, 283708289991521] [170800615664635332564330517881801, 5243626382156167474921782674260]
1062 - [306917, 9418]
1063 - [1745324831097224, 53531527412685]
1064 - [685, 21]
1065 - [82667999, 2533160]
1066 [3434907, 105205] [23597172197299, 722738781870]
1067 - [98, 3]
1068 - [11539207, 353094]
1069 [609477590160968572612782, 18640986378417772615285] [742925865816843150858935268959512942700219559049, 22722526912283010072320240710785462723519145740]
1070 - [2948491, 90138]
1071 - [28799, 880]
1072 - [4771081927, 145720107]
1073 [45368, 1385] [4116510849, 125669360]
1074 - [3489425, 106476]
1075 - [168867574226, 5150412729]
1076 - [13449, 410]
1077 - [23522399, 716760]
1078 - [197, 6]
1079 - [180576876, 5497325]
1080 - [5291, 161]
1081 - [293215056101715119615, 8918122218728007648]
1082 [1262101, 38369] [3185797868403, 96851106538]
1083 - [362, 11]
1084 - [26900393642040537919999, 817041270290926993200]
1085 - [17919, 544]
1086 - [725, 22]
1087 - [1088, 33]
1088 - [33, 1]
1090 [33, 1] [2179, 66]
1091 - [1090, 33]
1092 - [727, 22]
1093 [18018, 545] [649296649, 19639620]
1094 - [14898540581538085, 450438047467914]
1095 - [364, 11]
1096 - [3959299, 119595]
1097 [21019276, 634621] [883619927128353, 26678547908792]
1098 - [118097, 3564]
1099 - [1593163815326, 48057545715]
1100 - [199, 6]
1101 - [24313015, 732732]
1102 - [11382443, 342882]
1103 - [2346242745024, 70645611145]
1104 - [7775, 234]
1105 [28488, 857] [1623132289, 48828432]
1106 - [5055, 152]
1107 - [7263026, 218295]
1108 - [159150073798980475849, 4781200586939013510]
1109 [610206001542610, 18323595738617] [744702728637239515399291224201, 22362336179089374248695940740]
1110 - [1999, 60]
1111 - [100, 3]
1112 - [2501, 75]
1113 - [702463, 21056]
1114 [6111788103202293, 183115747156745] [74707907636890165020962560915699, 2238329289963186402088628832570]
1115 - [4560126, 136565]
1116 - [4620799, 138320]
1117 [209639866907824541118, 6272595595385543645] [87897747594260774254246835664214545379849, 2629972211566463612149241455626172190220]
1118 - [4213, 126]
1119 - [31657255, 946364]
1120 - [126001, 3765]
1121 - [151985104364951, 4539397558620]
1122 - [67, 2]
1123 - [16506645562632482, 492571123214799]
1124 - [2262200630049, 67475787740]
1125 - [1730726404001, 51600291864]
1126 - [91205, 2718]
1127 - [55224, 1645]
1128 - [2351, 70]
1129 [168, 5] [56449, 1680]
1130 [437, 13] [381939, 11362]
1131 - [368185, 10948]
1132 - [38239443505885447, 1136549486236362]
1133 - [514999, 15300]
1134 - [362074049, 10752040]
1135 - [10614754956124, 315073619505]
1136 - [24220799, 718620]
1137 - [650468934487, 19290626292]
1138 [348711, 10337] [243198723043, 7209251214]
1139 - [135, 4]
1140 - [2431, 72]
1141 - [1036782394157223963237125215, 30693385322765657197397208]
1142 - [906612101, 26828010]
1143 - [4730624, 139925]
1144 - [561835, 16611]
1145 [1252, 37] [3135009, 92648]
1146 - [3447551, 101840]
1147 - [9447152318, 278945403]
1148 - [165887, 4896]
1149 - [7207295, 212624]
1150 - [1184384449, 34925592]
1151 - [65905671840, 1942607807]
1152 - [577, 17]
1153 [102475040023072656, 3017890256875073] [21002267655460685387089654909788673, 618516849717028248841100388607776]
1154 - [1155, 34]
1155 - [34, 1]
1157 [34, 1] [2313, 68]
1158 - [1157, 34]
1159 - [1669145551424551, 49028938575180]
1160 - [579, 17]
1161 - [19153416854935, 562121440972]
1162 - [22616173, 663462]
1163 - [1260321002, 36956541]
1164 - [168199, 4930]
1165 [2959961778, 86720773] [17522747454441842569, 513380346877228788]
1166 - [660427701, 19340870]
1167 - [373828, 10943]
1168 - [2281249, 66750]
1169 - [184235994983935, 5388491828064]
1170 - [6499, 190]
1171 - [52268476130, 1527430263]
1172 - [12320649, 359890]
1173 - [137, 4]
1174 - [4672442879206569303591365, 136367209121366140674774]
1175 - [442224, 12901]
1176 - [4801, 140]
1177 - [9482010153904475351, 276383546065778460]
1178 - [59583, 1736]
1179 - [10610, 309]
1180 - [2024999, 58950]
1181 [12243764736718, 356278838045] [299819549856198391714823049, 8724388547388468773672620]
1182 - [27252587, 792682]
1183 - [130576328, 3796401]
1184 - [27365201, 795285]
1185 - [41418751, 1203200]
1186 - [6320195, 183522]
1187 - [39095175626, 1134743775]
1188 - [48599, 1410]
1189 [8476403117418, 245821958945] [143698819617947177389973449, 4167372038262395220808020]
1190 - [69, 2]
1191 - [3175, 92]
1192 - [335473872499, 9716739825]
1193 [854992268, 24753805] [1462023556679567649, 42328623757159480]
1194 - [21647468749, 626476750]
1195 - [4650060959, 134516232]
1196 - [415, 12]
1197 - [2588599, 74820]
1198 - [4206256963502359251599, 121525492983646316460]
1199 - [497777820, 14375599]
1200 - [1351, 39]
1201 [249013832327465057144832, 7185416856094423077385] [124015777380821763117401108566632174583648616449, 3578536376412875571438541467679723838177648640]
1202 - [10817, 312]
1203 - [434282, 12521]
1204 - [5883392537695, 169556554116]
1205 - [7174089, 206668]
1206 - [145925, 4202]
1207 - [1159928, 33387]
1208 - [4276623, 123046]
1209 - [57029335, 1640156]
1210 - [1499219281, 43099524]
1211 - [174, 5]
1212 - [12741151, 365980]
1213 [70600734, 2027117] [9968927282677513, 286231896207756]
1214 - [73150798374975, 2099470260104]
1215 - [244, 7]
1216 - [6681448801, 191603685]
1217 [27628256, 791969] [1526641059203073, 43761444552128]
1218 - [349, 10]
1219 - [26811112454855326, 767914700295135]
1220 - [489, 14]
1221 - [21385, 612]
1222 - [7734245409307, 221249524122]
1223 - [1224, 35]
1224 - [35, 1]
1226 [35, 1] [2451, 70]
1227 - [1226, 35]
1228 - [15674867542871047, 447305971699506]
1229 [21490, 613] [923640201, 26346740]
1230 - [491, 14]
1231 - [2051772574240109813487200, 58479034523501522751177]
1232 - [351, 10]
1233 - [74007554246657, 2107630689984]
1234 - [586327869067265, 16691023073856]
1235 - [246, 7]
1236 - [64202725495, 1826182722]
1237 [1083478144027661982, 30805999591330265] [2347849777171254083668416002496336649, 66755254524262858038831090292970460]
1238 - [902501, 25650]
1239 - [176, 5]
1240 - [848719, 24102]
1241 [33995032, 965005] [2311324401362049, 65610751710320]
1242 - [415151, 11780]
1243 - [171874, 4875]
1244 - [570130807708799, 16164576060360]
1245 - [7534241, 213528]
1246 - [443896693148735, 12575431287288]
1247 - [929188, 26313]
1248 - [5617, 159]
1249 [9235587034329029688240, 261326400283096392593] [170592135737332963602258146192126623188595201, 4827005428364886233636226185917486870412640]
1250 [275807, 7801] [152139002499, 4303140814]
1251 - [77563250, 2192943]
1252 - [32188120829134849, 909690079282080]
1253 - [2772351, 78320]
1254 - [362405, 10234]
1255 - [31134002572889, 878846634708]
1256 - [392499, 11075]
1257 - [101399, 2860]
1258 [114669, 3233] [26297959123, 741449754]
1259 - [111409143832810, 3139843281069]
1260 - [71, 2]
1261 [246622490957682, 6945046160725] [121645306092343879707029626249, 3425609167948170805090878900]
1262 - [61855669263, 1741204708]
1263 - [142297, 4004]
1264 - [12799, 360]
1265 - [206999, 5820]
1266 - [533240465, 14986708]
1267 - [625514462, 17573127]
1268 - [248678907849, 6983599490]
1269 - [3429215, 96264]
1270 - [581837113691, 16326743118]
1271 - [32799, 920]
1272 - [107, 3]
1273 - [183449440868134807, 5141646215851476]
1274 - [54100801, 1515720]
1275 - [4999, 140]
1276 - [332911854336799, 9319742702580]
1277 [144467948482, 4042742285] [41741976277195612208649, 1168093368310765922740]
1278 - [143, 4]
1279 - [2823156987094185989161256000, 78940478828277833993575041]
1280 - [51841, 1449]
1281 - [790329175, 22081748]
1282 - [3410326403, 95247138]
1283 - [1846514088226, 51551261505]
1284 - [215, 6]
1285 [8776850757318, 244842694225] [154066218432467100261106249, 4297895572464941510177100]
1286 - [25363110565, 707264778]
1287 - [287, 8]
1288 - [323, 9]
1289 [300125080421900, 8359413933853] [180150127796503885363999220001, 5017739558355166143865161400]
1290 - [431, 12]
1291 - [709367533957910186877445608410, 19742774580291798609489951267]
1292 - [647, 18]
1293 - [863, 24]
1294 - [1295, 36]
1295 - [36, 1]
1297 [36, 1] [2593, 72]
1298 - [1297, 36]
1299 - [865, 24]
1300 - [649, 18]
1301 [217318, 6025] [94454226249, 2618681900]
1302 - [433, 12]
1303 - [36556474232606087113228808, 1012726314518934688771461]
1304 - [325, 9]
1305 - [289, 8]
1306 [311761210150515, 8626815105389] [194390104309027154137909530451, 5379012634001634472555250670]
1307 - [195748082, 5414517]
1308 - [217, 6]
1309 - [803170174072265, 22199221738044]
1310 - [125100021, 3456382]
1311 - [193820, 5353]
1312 - [53137, 1467]
1313 [616, 17] [758913, 20944]
1314 - [145, 4]
1315 - [887511646, 24474351]
1316 - [2376415, 65508]
1317 - [37542877077093751, 1034509930681500]
1318 - [11393611468262768176276517, 313836679699540895794554]
1319 - [1213059240, 33401011]
1320 - [109, 3]
1321 [53450926715396612796432, 1470631386271961169925] [5714003133469398668515586785344107222143861249, 157213220905969449559233656682503745171415200]
1322 [29851, 821] [1782164403, 49015342]
1323 - [3650401, 100360]
1324 - [15519021083817312418194098721799, 426501271855353724503844374810]
1325 [182, 5] [66249, 1820]
1326 - [2549, 70]
1327 - [65024, 1785]
1328 - [13447, 369]
1329 - [704641456388201495, 19328835950934116]
1330 - [1246589, 34182]
1331 - [99612037019890, 2730378110727]
1332 - [73, 2]
1333 - [330126493332151, 9042016684140]
1334 - [3155972095, 86408256]
1335 - [45124, 1235]
1336 - [63804373719695, 1745609999622]
1337 - [1367431647, 37397272]
1338 - [17308813, 473194]
1339 - [150175, 4104]
1340 - [729631, 19932]
1341 - [1331459723602089239201, 36359143548676345320]
1342 - [1099, 30]
1343 - [388128, 10591]
1344 - [6049, 165]
1345 - [4841, 132]
1346 - [340550115, 9282362]
1347 - [6046682, 164753]
1348 - [2063810353129713793, 56211456782882376]
1349 - [2039180665951, 55520034840]
1350 - [470449, 12804]
1351 - [6175, 168]
1352 - [114243, 3107]
1353 - [42982433, 1168536]
1354 [13154959098165, 357503394613] [346105897748788220212734451, 9405885067178313198370290]
1355 - [1823289, 49532]
1356 - [19196241799, 521298370]
1357 - [355593479414372351, 9653034643064640]
1358 - [2999823, 81404]
1359 - [1728148040, 46878231]
1360 - [285769, 7749]
1361 [13256618645260, 359338458529] [351475875811710155440935201, 9527225818589057344845080]
1362 - [1370184257, 37127048]
1363 - [3451726, 93495]
1364 - [10626551, 287730]
1365 - [25271, 684]
1366 - [2291037052846189335142389540899525, 61987877112128467931286485364042]
1367 - [1368, 37]
1368 - [37, 1]
1370 [37, 1] [2739, 74]
1371 - [1370, 37]
1372 - [34100354867927167, 920623046934552]
1373 [25382, 685] [1288491849, 34773340]
1374 - [153544325045, 4142289074]
1375 - [89330852249, 2409073932]
1376 - [216528049, 5837205]
1377 - [41270070401, 1112162480]
1378 [42801, 1153] [3663851203, 98699106]
1379 - [2517968895, 67806016]
1380 - [6761, 182]
1381 [214347434092377992503819770876282, 5767949505076394676092612487905] [91889645003972654622127399341267476203024769394634584316300287049, 2472690352775053537868141555978544119564691977342423075952738420]
1382 - [13003237, 349782]
1383 - [1294948, 34821]
1384 - [17299, 465]
1385 [104092, 2797] [21670288929, 582290648]
1386 - [21295, 572]
1387 - [1510442, 40557]
1388 - [823306252807, 22098697686]
1389 - [625975, 16796]
1390 - [2298622285432981, 61653851607582]
1391 - [9081305, 243492]
1392 - [1567, 42]
1393 - [3583, 96]
1394 - [12545, 336]
1395 - [124001, 3320]
1396 - [169648201, 4540530]
1397 - [16033248351, 428966440]
1398 - [4866437, 130154]
1399 - [72267866478994045309640, 1932130144300442015349]
1400 - [449, 12]
1401 - [1653779660151229685015, 44183343633378497932]
1402 [930015700509, 24837980029] [1729858406386491965718163, 46199422791797974269522]
1403 - [74726, 1995]
1404 - [62425, 1666]
1405 - [2249, 60]
1406 - [75, 2]
1407 - [3751, 100]
1408 - [12048797377, 321101529]
1409 [35462524420653380, 944744019346217] [2515181276570874689623012210848801, 67006015714662899288959142526920]
1410 - [751, 20]
1411 - [444071330, 11821953]
1412 - [10157115393, 270304352]
1413 - [46698728731849, 1242321430740]
1414 - [182475063042299, 4852648311810]
1415 - [22029004, 585621]
1416 - [258065, 6858]
1417 [1640214636, 43572829] [5380608104297224993, 142937583715450488]
1418 [38899, 1033] [3026264403, 80365334]
1419 - [12770, 339]
1420 - [954809, 25338]
1421 - [9801, 260]
1422 - [119908097, 3179792]
1423 - [480011423186147603704208, 12724746017155626396759]
1424 - [500001, 13250]
1425 - [151, 4]
1426 - [9860975, 261132]
1427 - [9373918762, 248147097]
1428 - [3401, 90]
1429 [3375918, 89305] [22793644685449, 602972713980]
1430 - [13311, 352]
1431 - [1324, 35]
1432 - [176579805797, 4666266363]
1433 [5767329724, 152353237] [66524184290667832353, 1757342704595433176]
1434 - [176669098751, 4665369200]
1435 - [999046, 26373]
1436 - [259199, 6840]
1437 - [97359338210399, 2568320564040]
1438 - [15960728127743, 420894506064]
1439 - [12857985170030364960, 338955373761021521]
1440 - [721, 19]
1441 - [841014472843068524054999, 22154985959533110482700]
1442 - [1443, 38]
1443 - [38, 1]
1445 [38, 1] [2889, 76]
1446 - [1445, 38]
1447 - [130410331757421848, 3428291438910093]
1448 - [723, 19]
1449 - [277301249, 7284800]
1450 [3881493, 101933] [30131975818099, 791304451938]
1451 - [32507827844328530, 853403156620707]
1452 - [262087, 6878]
1453 [14325465250389344429699099982, 375816573334545637714354805] [410437909280225285392045277163541936816200282681664800649, 10767494523648864494064975684283690243920830620634227020]
1454 - [366707758095, 9616961884]
1455 - [23764, 623]
1456 - [4954951, 129855]
1457 - [7233162836703, 189495310312]
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1884 - [7838695, 180594]
1885 - [521, 12]
1886 - [92415, 2128]
1887 - [8992, 207]
1888 - [188396089777, 4335816459]
1889 [4300062425132968, 98937010028785] [36981073720080844052144960978049, 850870638559564006506264967760]
1890 - [97469, 2242]
1891 - [819136267782775550, 18836941567957233]
1892 - [87, 2]
1893 - [5047, 116]
1894 - [226309103832669021290704832069984034245, 5200104096794640672678934313809609086]
1895 - [54849639, 1259996]
1896 - [193549, 4445]
1897 - [43903, 1008]
1898 [2309, 53] [10662963, 244754]
1899 - [278354373650, 6387568451]
1900 - [57799, 1326]
1901 [218, 5] [95049, 2180]
1902 - [2846027, 65258]
1903 - [875034172, 20058831]
1904 - [28799, 660]
1905 - [10709910761, 245379548]
1906 [10705292457, 245209385] [229206573179802193699, 5250076359252217890]
1907 - [17162, 393]
1908 - [8777860001, 200955300]
1909 - [2380667255, 54487356]
1910 - [278479, 6372]
1911 - [62425, 1428]
1912 - [1617319577991743, 36987237828948]
1913 [4748651845484, 108570751717] [45099388699237198022388513, 1031129401013034423392056]
1914 - [175, 4]
1915 - [1027150205228001406, 23471971442429307]
1916 - [17873503027199, 408330599040]
1917 - [194399, 4440]
1918 - [416753, 9516]
1919 - [77244401049, 1763313460]
1920 - [116161, 2651]
1921 [71388815998932, 1628795259625] [10192726099458738978050281249, 232555530178603601325441000]
1922 - [152139002499, 3470274850]
1923 - [4367227226, 99590095]
1924 - [1859131879201, 42384558540]
1925 - [351, 8]
1926 - [695359189925, 15844591882]
1927 - [46636702417448, 1062397346883]
1928 - [483, 11]
1929 - [126794455, 2886916]
1930 [8020227, 182561] [128648082263059, 2928361322694]
1931 - [53405823410, 1215339123]
1932 - [967, 22]
1933 [268143549167208172282776882, 6098898790460359344402565] [143801925919973973006746044614830462252244345987283849, 3270760735371267636227864743687616171541802967004660]
1934 - [1935, 44]
1935 - [44, 1]
1937 [44, 1] [3873, 88]
1938 - [1937, 44]
1939 - [6672165474011915935, 151522771012903896]
1940 - [969, 22]
1941 - [582709607499783953373016131935, 13226331726483194685799879992]
1942 - [6984160935186866597, 158485533243485118]
1943 - [83751, 1900]
1944 - [485, 11]
1945 - [2632285113641, 59686089372]
1946 - [121257485, 2748762]
1947 - [353, 8]
1948 - [5388481003084995290965125247, 122087690908458416096046936]
1949 [83218, 1885] [13850471049, 313731860]
1950 - [595349, 13482]
1951 - [226742438218831488120936558859828640, 5133389024333877559662473405561007]
1952 - [118097, 2673]
1953 - [3844063, 86984]
1954 - [4691555, 106134]
1955 - [2874, 65]
1956 - [7592629975, 171675298]
1957 - [400881151, 9061920]
1958 - [177, 4]
1959 - [9719730001876, 219602349925]
1960 - [1039681, 23484]
1961 - [5431118185, 122645172]
1962 - [31753512017, 716873068]
1963 - [24445564472798, 551746923459]
1964 - [17532421310198941977799, 395613548729114914110]
1965 - [1176119, 26532]
1966 - [9850975268107008669640939861775, 222171052269092942126897458308]
1967 - [2528, 57]
1968 - [29767, 671]
1969 - [46885009158085926066844328473751, 1056601315610203746069441327300]
1970 [577, 13] [665859, 15002]
1971 - [1620674, 36505]
1972 - [935662752649, 21070065510]
1973 [88526, 1993] [15673705353, 352864636]
1974 - [1376255, 30976]
1975 - [12799, 288]
1976 - [73035, 1643]
1977 - [230201397518423, 5177315380292]
1978 - [60973994702573, 1370981217774]
1979 - [196956439969690, 4427384992341]
1980 - [89, 2]
1981 - [478731918732725530377313576551512366455, 10755983977504786513343290157817764652]
1982 - [438784358463, 9855968632]
1983 - [354724327, 7965804]
1984 - [4620799, 103740]
1985 [1907602928, 42816161] [7277897861828346369, 163352468178638816]
1986 - [13209364625, 296409628]
1987 - [2163842, 48543]
1988 - [1201887, 26956]
1989 - [5544449, 124320]
1990 - [35314095433681999, 791628731408580]
1991 - [95638951, 2143380]
1992 - [179777, 4028]
1993 [6911348232395343251232, 154813605922550351605] [95533468778868471170054252659885376539035649, 2139941483287135231593131558297490898854720]
1994 [84198805, 1885573] [14178877526856051, 317525986680530]
1995 - [134, 3]
1996 - [40320199, 902490]
1997 [384413533882, 8602202005] [295547530063295123979849, 6613605743817751666820]
1998 - [13231349, 296010]
1999 - [4027701399389138208695911951306886478800, 90084665203202024260494303744425250249]

[参考文献]


Last Update: 2005.06.12
H.Nakao

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