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Integral Points on Conic Curves: x^2-Dy^2=±1(D \in [2,999])


[2003.08.16]x2-Dy2=±1 (D \in [2,999])の整点


■2≦D≦9999かつ平方数でない整数Dについて、Pell方程式
     x2-Dy2 = ±1
の基本解を連分数を使って求める。

pari/gpで計算すると、以下のようになる。
gp> read("pell.gp")
time = 42 ms.
gp> findpell(1,10000,50)
   realprecision = 57 significant digits (50 digits displayed)
d=2; [1, 1, -1]
d=2; [3, 2, 1]
d=3; [2, 1, 1]
d=5; [2, 1, -1]
d=5; [9, 4, 1]
d=6; [5, 2, 1]
...省略...
   realprecision = 105 significant digits (100 digits displayed)
d=9997; [1056881013819720204460013572819482, 10570395816504471539188482748085, -1]
d=9997; [2233994954745199219608947478540746520621707535213601737381917496649, 22343301294045950038591020734822406588061732170848426457972383940, 1]
   realprecision = 57 significant digits (50 digits displayed)
d=9998; [9999, 100, 1]
d=9999; [100, 1, 1]
time = 36,687 ms.
gp> quit
Good bye!
bash-2.05a$ 

■2≦D≦9999について、x2-Dy2=1の基本解が特に大きいのは、D=9949,9349の場合であり、それぞれ、以下のようになる。

■2≦D≦999についての結果は、以下の通りである。
D fundamental solution [x,y] of x2-Dy2=-1 fundamental solution [x,y] of x2-Dy2=1
2 [1, 1] [3, 2]
3 - [2, 1]
5 [2, 1] [9, 4]
6 - [5, 2]
7 - [8, 3]
8 - [3, 1]
10 [3, 1] [19, 6]
11 - [10, 3]
12 - [7, 2]
13 [18, 5] [649, 180]
14 - [15, 4]
15 - [4, 1]
17 [4, 1] [33, 8]
18 - [17, 4]
19 - [170, 39]
20 - [9, 2]
21 - [55, 12]
22 - [197, 42]
23 - [24, 5]
24 - [5, 1]
26 [5, 1] [51, 10]
27 - [26, 5]
28 - [127, 24]
29 [70, 13] [9801, 1820]
30 - [11, 2]
31 - [1520, 273]
32 - [17, 3]
33 - [23, 4]
34 - [35, 6]
35 - [6, 1]
37 [6, 1] [73, 12]
38 - [37, 6]
39 - [25, 4]
40 - [19, 3]
41 [32, 5] [2049, 320]
42 - [13, 2]
43 - [3482, 531]
44 - [199, 30]
45 - [161, 24]
46 - [24335, 3588]
47 - [48, 7]
48 - [7, 1]
50 [7, 1] [99, 14]
51 - [50, 7]
52 - [649, 90]
53 [182, 25] [66249, 9100]
54 - [485, 66]
55 - [89, 12]
56 - [15, 2]
57 - [151, 20]
58 [99, 13] [19603, 2574]
59 - [530, 69]
60 - [31, 4]
61 [29718, 3805] [1766319049, 226153980]
62 - [63, 8]
63 - [8, 1]
65 [8, 1] [129, 16]
66 - [65, 8]
67 - [48842, 5967]
68 - [33, 4]
69 - [7775, 936]
70 - [251, 30]
71 - [3480, 413]
72 - [17, 2]
73 [1068, 125] [2281249, 267000]
74 [43, 5] [3699, 430]
75 - [26, 3]
76 - [57799, 6630]
77 - [351, 40]
78 - [53, 6]
79 - [80, 9]
80 - [9, 1]
82 [9, 1] [163, 18]
83 - [82, 9]
84 - [55, 6]
85 [378, 41] [285769, 30996]
86 - [10405, 1122]
87 - [28, 3]
88 - [197, 21]
89 [500, 53] [500001, 53000]
90 - [19, 2]
91 - [1574, 165]
92 - [1151, 120]
93 - [12151, 1260]
94 - [2143295, 221064]
95 - [39, 4]
96 - [49, 5]
97 [5604, 569] [62809633, 6377352]
98 - [99, 10]
99 - [10, 1]
101 [10, 1] [201, 20]
102 - [101, 10]
103 - [227528, 22419]
104 - [51, 5]
105 - [41, 4]
106 [4005, 389] [32080051, 3115890]
107 - [962, 93]
108 - [1351, 130]
109 [8890182, 851525] [158070671986249, 15140424455100]
110 - [21, 2]
111 - [295, 28]
112 - [127, 12]
113 [776, 73] [1204353, 113296]
114 - [1025, 96]
115 - [1126, 105]
116 - [9801, 910]
117 - [649, 60]
118 - [306917, 28254]
119 - [120, 11]
120 - [11, 1]
122 [11, 1] [243, 22]
123 - [122, 11]
124 - [4620799, 414960]
125 [682, 61] [930249, 83204]
126 - [449, 40]
127 - [4730624, 419775]
128 - [577, 51]
129 - [16855, 1484]
130 [57, 5] [6499, 570]
131 - [10610, 927]
132 - [23, 2]
133 - [2588599, 224460]
134 - [145925, 12606]
135 - [244, 21]
136 - [35, 3]
137 [1744, 149] [6083073, 519712]
138 - [47, 4]
139 - [77563250, 6578829]
140 - [71, 6]
141 - [95, 8]
142 - [143, 12]
143 - [12, 1]
145 [12, 1] [289, 24]
146 - [145, 12]
147 - [97, 8]
148 - [73, 6]
149 [113582, 9305] [25801741449, 2113761020]
150 - [49, 4]
151 - [1728148040, 140634693]
152 - [37, 3]
153 - [2177, 176]
154 - [21295, 1716]
155 - [249, 20]
156 - [25, 2]
157 [4832118, 385645] [46698728731849, 3726964292220]
158 - [7743, 616]
159 - [1324, 105]
160 - [721, 57]
161 - [11775, 928]
162 - [19601, 1540]
163 - [64080026, 5019135]
164 - [2049, 160]
165 - [1079, 84]
166 - [1700902565, 132015642]
167 - [168, 13]
168 - [13, 1]
170 [13, 1] [339, 26]
171 - [170, 13]
172 - [24248647, 1848942]
173 [1118, 85] [2499849, 190060]
174 - [1451, 110]
175 - [2024, 153]
176 - [199, 15]
177 - [62423, 4692]
178 - [1601, 120]
179 - [4190210, 313191]
180 - [161, 12]
181 [1111225770, 82596761] [2469645423824185801, 183567298683461940]
182 - [27, 2]
183 - [487, 36]
184 - [24335, 1794]
185 [68, 5] [9249, 680]
186 - [7501, 550]
187 - [1682, 123]
188 - [4607, 336]
189 - [55, 4]
190 - [52021, 3774]
191 - [8994000, 650783]
192 - [97, 7]
193 [1764132, 126985] [6224323426849, 448036604040]
194 - [195, 14]
195 - [14, 1]
197 [14, 1] [393, 28]
198 - [197, 14]
199 - [16266196520, 1153080099]
200 - [99, 7]
201 - [515095, 36332]
202 [3141, 221] [19731763, 1388322]
203 - [57, 4]
204 - [4999, 350]
205 - [39689, 2772]
206 - [59535, 4148]
207 - [1151, 80]
208 - [649, 45]
209 - [46551, 3220]
210 - [29, 2]
211 - [278354373650, 19162705353]
212 - [66249, 4550]
213 - [194399, 13320]
214 - [695359189925, 47533775646]
215 - [44, 3]
216 - [485, 33]
217 - [3844063, 260952]
218 [251, 17] [126003, 8534]
219 - [74, 5]
220 - [89, 6]
221 - [1665, 112]
222 - [149, 10]
223 - [224, 15]
224 - [15, 1]
226 [15, 1] [451, 30]
227 - [226, 15]
228 - [151, 10]
229 [1710, 113] [5848201, 386460]
230 - [91, 6]
231 - [76, 5]
232 - [19603, 1287]
233 [23156, 1517] [1072400673, 70255304]
234 - [5201, 340]
235 - [46, 3]
236 - [561799, 36570]
237 - [228151, 14820]
238 - [11663, 756]
239 - [6195120, 400729]
240 - [31, 2]
241 [71011068, 4574225] [10085143557001249, 649641205044600]
242 - [19601, 1260]
243 - [70226, 4505]
244 - [1766319049, 113076990]
245 - [51841, 3312]
246 - [88805, 5662]
247 - [85292, 5427]
248 - [63, 4]
249 - [8553815, 542076]
250 [4443, 281] [39480499, 2496966]
251 - [3674890, 231957]
252 - [127, 8]
253 - [3222617399, 202604220]
254 - [255, 16]
255 - [16, 1]
257 [16, 1] [513, 32]
258 - [257, 16]
259 - [847225, 52644]
260 - [129, 8]
261 - [192119201, 11891880]
262 - [104980517, 6485718]
263 - [139128, 8579]
264 - [65, 4]
265 [6072, 373] [73738369, 4529712]
266 - [685, 42]
267 - [2402, 147]
268 - [4771081927, 291440214]
269 [82, 5] [13449, 820]
270 - [5291, 322]
271 - [115974983600, 7044978537]
272 - [33, 2]
273 - [727, 44]
274 [1407, 85] [3959299, 239190]
275 - [199, 12]
276 - [7775, 468]
277 [8920484118, 535979945] [159150073798980475849, 9562401173878027020]
278 - [2501, 150]
279 - [1520, 91]
280 - [251, 15]
281 [1063532, 63445] [2262200630049, 134951575480]
282 - [2351, 140]
283 - [138274082, 8219541]
284 - [24220799, 1437240]
285 - [2431, 144]
286 - [561835, 33222]
287 - [288, 17]
288 - [17, 1]
290 [17, 1] [579, 34]
291 - [290, 17]
292 - [2281249, 133500]
293 [2482, 145] [12320649, 719780]
294 - [4801, 280]
295 - [2024999, 117900]
296 - [3699, 215]
297 - [48599, 2820]
298 [409557, 23725] [335473872499, 19433479650]
299 - [415, 24]
300 - [1351, 78]
301 - [5883392537695, 339113108232]
302 - [4276623, 246092]
303 - [2524, 145]
304 - [57799, 3315]
305 - [489, 28]
306 - [35, 2]
307 - [88529282, 5052633]
308 - [351, 20]
309 - [64202725495, 3652365444]
310 - [848719, 48204]
311 - [16883880, 957397]
312 - [53, 3]
313 [126862368, 7170685] [32188120829134849, 1819380158564160]
314 [443, 25] [392499, 22150]
315 - [71, 4]
316 - [12799, 720]
317 [352618, 19805] [248678907849, 13967198980]
318 - [107, 6]
319 - [12901780, 722361]
320 - [161, 9]
321 - [215, 12]
322 - [323, 18]
323 - [18, 1]
325 [18, 1] [649, 36]
326 - [325, 18]
327 - [217, 12]
328 - [163, 9]
329 - [2376415, 131016]
330 - [109, 6]
331 - [2785589801443970, 153109862634573]
332 - [13447, 738]
333 - [73, 4]
334 - [63804373719695, 3491219999244]
335 - [604, 33]
336 - [55, 3]
337 [1015827336, 55335641] [2063810353129713793, 112422913565764752]
338 [239, 13] [114243, 6214]
339 - [97970, 5321]
340 - [285769, 15498]
341 - [10626551, 575460]
342 - [37, 2]
343 - [130576328, 7050459]
344 - [10405, 561]
345 - [6761, 364]
346 [93, 5] [17299, 930]
347 - [641602, 34443]
348 - [1567, 84]
349 [9210, 493] [169648201, 9081060]
350 - [449, 24]
351 - [62425, 3332]
352 - [77617, 4137]
353 [71264, 3793] [10157115393, 540608704]
354 - [258065, 13716]
355 - [954809, 50676]
356 - [500001, 26500]
357 - [3401, 180]
358 - [176579805797, 9332532726]
359 - [360, 19]
360 - [19, 1]
362 [19, 1] [723, 38]
363 - [362, 19]
364 - [4954951, 259710]
365 [3458, 181] [23915529, 1251796]
366 - [907925, 47458]
367 - [19019995568, 992835687]
368 - [1151, 60]
369 - [8396801, 437120]
370 [327, 17] [213859, 11118]
371 - [1695, 88]
372 - [12151, 630]
373 [5118, 265] [52387849, 2712540]
374 - [3365, 174]
375 - [15124, 781]
376 - [2143295, 110532]
377 - [233, 12]
378 - [8749, 450]
379 - [12941197220540690, 664744650125541]
380 - [39, 2]
381 - [1015, 52]
382 - [164998439999, 8442054600]
383 - [18768, 959]
384 - [4801, 245]
385 - [95831, 4884]
386 - [111555, 5678]
387 - [3482, 177]
388 - [62809633, 3188676]
389 [1282, 65] [3287049, 166660]
390 - [79, 4]
391 - [7338680, 371133]
392 - [99, 5]
393 - [46437143, 2342444]
394 [395023035, 19900973] [312086396361222451, 15722685507826110]
395 - [159, 8]
396 - [199, 10]
397 [20478302982, 1027776565] [838721786045180184649, 42094239791738433660]
398 - [399, 20]
399 - [20, 1]
401 [20, 1] [801, 40]
402 - [401, 20]
403 - [669878, 33369]
404 - [201, 10]
405 - [161, 8]
406 - [59468095, 2951352]
407 - [2663, 132]
408 - [101, 5]
409 [111921796968, 5534176685] [25052977273092427986049, 1238789998647218582160]
410 - [81, 4]
411 - [49730, 2453]
412 - [103537981567, 5100950232]
413 - [113399, 5580]
414 - [24335, 1196]
415 - [18412804, 903849]
416 - [5201, 255]
417 - [85322647, 4178268]
418 - [33857, 1656]
419 - [270174970, 13198911]
420 - [41, 2]
421 [44042445696821418, 2146497463530785] [3879474045914926879468217167061449, 189073995951839020880499780706260]
422 - [7022501, 341850]
423 - [4607, 224]
424 - [32080051, 1557945]
425 [268, 13] [143649, 6968]
426 - [88751, 4300]
427 - [62, 3]
428 - [1850887, 89466]
429 - [1524095, 73584]
430 - [2862251, 138030]
431 - [151560720, 7300423]
432 - [1351, 65]
433 [7230660684, 347483377] [104564907854286695713, 5025068784834899736]
434 - [125, 6]
435 - [146, 7]
436 - [158070671986249, 7570212227550]
437 - [4599, 220]
438 - [293, 14]
439 - [440, 21]
440 - [21, 1]
442 [21, 1] [883, 42]
443 - [442, 21]
444 - [295, 14]
445 [4662, 221] [43468489, 2060604]
446 - [110166015, 5216512]
447 - [148, 7]
448 - [127, 6]
449 [189471332, 8941705] [71798771299708449, 3388393513402120]
450 - [19601, 924]
451 - [46471490, 2188257]
452 - [1204353, 56648]
453 - [1653751, 77700]
454 - [16916040084175685, 793909098494766]
455 - [64, 3]
456 - [1025, 48]
457 [59089951584, 2764111349] [6983244756398928218113, 326662411570389853632]
458 [107, 5] [22899, 1070]
459 - [499850, 23331]
460 - [2535751, 118230]
461 [24314110, 1132421] [1182351890184201, 55067617520620]
462 - [43, 2]
463 - [247512720456368, 11502891625161]
464 - [9801, 455]
465 - [15871, 736]
466 - [938319425, 43466808]
467 - [1625626, 75225]
468 - [649, 30]
469 - [137215, 6336]
470 - [1691, 78]
471 - [7838695, 361188]
472 - [306917, 14127]
473 - [87, 4]
474 - [193549, 8890]
475 - [57799, 2652]
476 - [28799, 1320]
477 - [8777860001, 401910600]
478 - [1617319577991743, 73974475657896]
479 - [2989440, 136591]
480 - [241, 11]
481 [964140, 43961] [1859131879201, 84769117080]
482 - [483, 22]
483 - [22, 1]
485 [22, 1] [969, 44]
486 - [485, 22]
487 - [51906073840568, 2352088722477]
488 - [243, 11]
489 - [7592629975, 343350596]
490 - [1039681, 46968]
491 - [93628044170, 4225374483]
492 - [29767, 1342]
493 [683982, 30805] [935662752649, 42140131020]
494 - [73035, 3286]
495 - [89, 4]
496 - [4620799, 207480]
497 - [1201887, 53912]
498 - [179777, 8056]
499 - [4490, 201]
500 - [930249, 41602]
501 - [11242731902975, 502288218432]
502 - [3832352837, 171046278]
503 - [24648, 1099]
504 - [449, 20]
505 - [809, 36]
506 - [45, 2]
507 - [1351, 60]
508 - [44757606858751, 1985797689600]
509 [395727950, 17540333] [313201220822405001, 13882400040814700]
510 - [271, 12]
511 - [4188548960, 185290497]
512 - [665857, 29427]
513 - [13771351, 608020]
514 - [4625, 204]
515 - [17406, 767]
516 - [16855, 742]
517 - [590968985399, 25990786260]
518 - [2367, 104]
519 - [14851876, 651925]
520 - [6499, 285]
521 [128377240, 5624309] [32961431500035201, 1444066532654320]
522 - [19603, 858]
523 - [81810300626, 3577314675]
524 - [225144199, 9835470]
525 - [6049, 264]
526 - [84056091546952933775, 3665019757324295532]
527 - [528, 23]
528 - [23, 1]
530 [23, 1] [1059, 46]
531 - [530, 23]
532 - [2588599, 112230]
533 [6118, 265] [74859849, 3242540]
534 - [3678725, 159194]
535 - [1618804, 69987]
536 - [145925, 6303]
537 - [192349463, 8300492]
538 [69051, 2977] [9536081203, 411129654]
539 - [3970, 171]
540 - [119071, 5124]
541 [1361516316469227450, 58536158470221581] [3707453360023867028800645599667005001, 159395869721270110077187138775196900]
542 - [4293183, 184408]
543 - [669337, 28724]
544 - [2449, 105]
545 - [1961, 84]
546 - [701, 30]
547 - [160177601264642, 6848699678673]
548 - [6083073, 259856]
549 - [1766319049, 75384660]
550 - [30580901, 1303974]
551 - [8380, 357]
552 - [47, 2]
553 - [624635837407, 26562217704]
554 [174293, 7405] [60756099699, 2581279330]
555 - [1814, 77]
556 - [12032115501124999, 510275358434250]
557 [118, 5] [27849, 1180]
558 - [7937, 336]
559 - [506568295, 21425556]
560 - [71, 3]
561 - [522785, 22072]
562 - [220938497, 9319728]
563 - [68122, 2871]
564 - [95, 4]
565 [14752278, 620633] [435259412378569, 18311501103948]
566 - [95609285, 4018758]
567 - [2024, 85]
568 - [143, 6]
569 [2894863832, 121359005] [16760473211643448449, 702635588524014320]
570 - [191, 8]
571 - [181124355061630786130, 7579818350628982587]
572 - [287, 12]
573 - [383, 16]
574 - [575, 24]
575 - [24, 1]
577 [24, 1] [1153, 48]
578 - [577, 24]
579 - [385, 16]
580 - [289, 12]
581 - [152071153975, 6308974548]
582 - [193, 8]
583 - [8429543, 349116]
584 - [145, 6]
585 - [33281, 1376]
586 [4115086707, 169992665] [33867877212256207699, 1399069112058008310]
587 - [1907162, 78717]
588 - [97, 4]
589 - [41423166067036218751, 1706811823063746000]
590 - [5781, 238]
591 - [165676, 6815]
592 - [73, 3]
593 [600632, 24665] [721517598849, 29629176560]
594 - [1098305, 45064]
595 - [18514, 759]
596 - [25801741449, 1056880510]
597 - [463287093751, 18961078500]
598 - [1574351, 64380]
599 - [24686379794520, 1008658133851]
600 - [49, 2]
601 [139468303679532, 5689030769845] [38902815462492318420311478049, 1586878942101888360258625080]
602 - [687, 28]
603 - [48842, 1989]
604 - [5972991296311683199, 243037569063951720]
605 - [930249, 37820]
606 - [42187499, 1713750]
607 - [164076033968, 6659640783]
608 - [2737, 111]
609 - [605695, 24544]
610 [71847, 2909] [10323982819, 418005846]
611 - [236926, 9585]
612 - [2177, 88]
613 [481673579088618, 19454612624065] [464018873584078278910994299849, 18741545784831997880308784340]
614 - [348291186245, 14055888354]
615 - [124, 5]
616 - [21295, 858]
617 [41009716, 1650989] [3363593612801313, 135413180018248]
618 - [10093, 406]
619 - [517213510553282930, 20788566180548739]
620 - [249, 10]
621 - [7775, 312]
622 - [13804370063, 553504812]
623 - [624, 25]
624 - [25, 1]
626 [25, 1] [1251, 50]
627 - [626, 25]
628 - [46698728731849, 1863482146110]
629 [7850, 313] [123245001, 4914100]
630 - [251, 10]
631 - [48961575312998650035560, 1949129537575151036427]
632 - [7743, 308]
633 - [440772247, 17519124]
634 [65999458125, 2621173333] [8711856945587257031251, 345992039259400361250]
635 - [126, 5]
636 - [3505951, 139020]
637 - [1419278889601, 56233877040]
638 - [42283, 1674]
639 - [24220799, 958160]
640 - [1039681, 41097]
641 [36120833468, 1426687145] [2609429220845977814049, 103066257550962737720]
642 - [5777, 228]
643 - [1988960193026, 78436933185]
644 - [11775, 464]
645 - [1024001, 40320]
646 - [305, 12]
647 - [120187368, 4725053]
648 - [19601, 770]
649 - [1123593226162199, 44104892095380]
650 - [51, 2]
651 - [1735, 68]
652 - [8212499464321351, 321626301297510]
653 [2291286382, 89664965] [10499986568677299849, 410896226494013260]
654 - [8915765, 348634]
655 - [737709209, 28824684]
656 - [2049, 80]
657 - [2281249, 89000]
658 - [1693, 66]
659 - [5930, 231]
660 - [1079, 42]
661 [2865454435422583218, 111453260296346905] [16421658242965910275055840472270471049, 638728478116949861246791167518480580]
662 - [1718102501, 66775950]
663 - [103, 4]
664 - [1700902565, 66007821]
665 - [13719, 532]
666 - [27365201, 1060380]
667 - [107119097, 4147668]
668 - [56447, 2184]
669 - [14226117859054135, 550013492618436]
670 - [5791211, 223734]
671 - [58620, 2263]
672 - [337, 13]
673 [48813455293932, 1881620424025] [4765506835465395993032041249, 183696788896587421699032600]
674 - [675, 26]
675 - [26, 1]
677 [26, 1] [1353, 52]
678 - [677, 26]
679 - [17792625320, 682818291]
680 - [339, 13]
681 - [10743166003415, 411679015748]
682 - [1197901, 45870]
683 - [170067682, 6507459]
684 - [57799, 2210]
685 [218623878, 8353189] [95592800063517769, 3652413145693884]
686 - [10850138895, 414260228]
687 - [165337, 6308]
688 - [24248647, 924471]
689 - [105, 4]
690 - [1471, 56]
691 - [31138100617500578690, 1184549173291009383]
692 - [2499849, 95030]
693 - [246401, 9360]
694 - [38782105445014642382885, 1472148590903997672114]
695 - [33639, 1276]
696 - [1451, 55]
697 [132, 5] [34849, 1320]
698 [5099, 193] [51999603, 1968214]
699 - [2271050, 85899]
700 - [8193151, 309672]
701 [11782, 445] [277631049, 10485980]
702 - [53, 2]
703 - [1159172, 43719]
704 - [79201, 2985]
705 - [237161, 8932]
706 - [34595, 1302]
707 - [2526, 95]
708 - [62423, 2346]
709 [18245310, 685217] [665782673992201, 25003993164540]
710 - [1279, 48]
711 - [80, 3]
712 - [1601, 60]
713 - [5286367, 197976]
714 - [4115, 154]
715 - [75646, 2829]
716 - [35115719688199, 1312336060110]
717 - [6998399, 261360]
718 - [8933399183036079503, 333391496474140716]
719 - [403480310400, 15047276489]
720 - [161, 6]
721 - [18632176943292415, 693898530122112]
722 - [22619537, 841812]
723 - [242, 9]
724 - [2469645423824185801, 91783649341730970]
725 - [9801, 364]
726 - [485, 18]
727 - [728, 27]
728 - [27, 1]
730 [27, 1] [1459, 54]
731 - [730, 27]
732 - [487, 18]
733 [9882, 365] [195307849, 7213860]
734 - [10394175, 383656]
735 - [244, 9]
736 - [24335, 897]
737 - [252975383, 9318468]
738 - [163, 6]
739 - [98015661073616742153890, 3605564376516452758671]
740 - [9249, 340]
741 - [7352695, 270108]
742 - [263091151, 9658380]
743 - [714024, 26195]
744 - [7501, 275]
745 - [12769001, 467820]
746 [5534843, 202645] [61268974069299, 2243216519470]
747 - [82, 3]
748 - [5658247, 206886]
749 - [1084616384895, 39631020176]
750 - [2550251, 93122]
751 - [7293318466794882424418960, 266136970677206024456793]
752 - [4607, 168]
753 - [308526027863, 11243313484]
754 [20457, 745] [836977699, 30480930]
755 - [1209, 44]
756 - [55, 2]
757 [1369326, 49769] [3750107388553, 136299971388]
758 - [413959717, 15035694]
759 - [551, 20]
760 - [52021, 1887]
761 [800, 29] [1280001, 46400]
762 - [6349, 230]
763 - [719724601, 26055780]
764 - [161784071999999, 5853142302000]
765 - [285769, 10332]
766 - [145933611945744638015, 5272795728865625208]
767 - [31212, 1127]
768 - [18817, 679]
769 [16367374077549540, 590222604844777] [535781868388881310859702308423201, 19320788325040337217824455505160]
770 - [111, 4]
771 - [2989136930, 107651137]
772 - [6224323426849, 224018302020]
773 [1343018, 48305] [3607394696649, 129748968980]
774 - [10405, 374]
775 - [4620799, 165984]
776 - [195, 7]
777 - [223, 8]
778 [54610269, 1957873] [5964562960504723, 213839942395674]
779 - [11785490, 422259]
780 - [391, 14]
781 - [67606199, 2419140]
782 - [783, 28]
783 - [28, 1]
785 [28, 1] [1569, 56]
786 - [785, 28]
787 - [34625394242, 1234262007]
788 - [393, 14]
789 - [16116667272575, 573768548496]
790 - [6616066879, 235389096]
791 - [225, 8]
792 - [197, 7]
793 - [4393, 156]
794 [30235, 1073] [1828310451, 64884310]
795 - [6626, 235]
796 - [529178298454520220799, 18756227493635055480]
797 [24715982, 875485] [1221759532448649, 43276943002540]
798 - [113, 4]
799 - [424, 15]
800 - [19601, 693]
801 - [500002000001, 17666702000]
802 - [295496099, 10434330]
803 - [7226, 255]
804 - [515095, 18166]
805 - [1514868641, 53392104]
806 - [6166395, 217202]
807 - [51841948, 1824923]
808 - [19731763, 694161]
809 [433852026040, 15253424933] [376455160998025676163201, 13235458622462202510640]
810 - [27379, 962]
811 - [1382072163578616410, 48531117622921197]
812 - [57, 2]
813 - [2167, 76]
814 - [4206992174549, 147454999410]
815 - [156644, 5487]
816 - [4999, 175]
817 - [343, 12]
818 [143, 5] [40899, 1430]
819 - [1574, 55]
820 - [39689, 1386]
821 [2121436703918, 74038651465] [9000987377460935993101449, 314136625452886403879740]
822 - [7397, 258]
823 - [235170474903644006168, 8197527430497636651]
824 - [59535, 2074]
825 - [48599, 1692]
826 - [222239304685, 7732694382]
827 - [900602, 31317]
828 - [1151, 40]
829 [15489282, 537965] [479835713751049, 16665383182260]
830 - [146411, 5082]
831 - [9799705, 339948]
832 - [842401, 29205]
833 - [9478657, 328416]
834 - [6552578705, 226897244]
835 - [34336355806, 1188258591]
836 - [46551, 1610]
837 - [12151, 420]
838 - [42112785797, 1454762046]
839 - [840, 29]
840 - [29, 1]
842 [29, 1] [1683, 58]
843 - [842, 29]
844 - [154962314660167628644999, 5334022845973817148450]
845 [12238, 421] [299537289, 10304396]
846 - [2143295, 73688]
847 - [8193151, 281520]
848 - [66249, 2275]
849 - [1501654712948695, 51536656330476]
850 - [2449, 84]
851 - [8418574, 288585]
852 - [194399, 6660]
853 [10379165085018, 355375843945] [215454135724113414336120649, 7377009103065498851032020]
854 - [1294299, 44290]
855 - [3041, 104]
856 - [695359189925, 23766887823]
857 [8118568, 277325] [131822292741249, 4502963741200]
858 - [703, 24]
859 - [2058844771979643060124010, 70246877103894937291269]
860 - [3871, 132]
861 - [541601801, 18457740]
862 - [358022566147312125503, 12194296994921665128]
863 - [18524026608, 630565199]
864 - [470449, 16005]
865 [348345108, 11844089] [242688628535063329, 8251660923733224]
866 - [42435, 1442]
867 - [70226, 2385]
868 - [3844063, 130476]
869 - [60192738698751, 2041898807200]
870 - [59, 2]
871 - [19442812076, 658794555]
872 - [126003, 4267]
873 - [62809633, 2125784]
874 - [3725, 126]
875 - [120126, 4061]
876 - [10951, 370]
877 [241326, 8149] [116476476553, 3933131148]
878 - [9314703, 314356]
879 - [107245324, 3617295]
880 - [89, 3]
881 [106316171432, 3581882825] [22606256615916825861249, 761624136944072910800]
882 - [19601, 660]
883 - [34878475759617272473442, 1173754162936357802169]
884 - [1665, 56]
885 - [119, 4]
886 - [7743524593057655851637765, 260148796464024194850378]
887 - [469224, 15755]
888 - [149, 5]
889 - [13231974717803657215, 443786188413453504]
890 - [179, 6]
891 - [3970, 133]
892 - [100351, 3360]
893 - [6091434999, 203842100]
894 - [299, 10]
895 - [359, 12]
896 - [449, 15]
897 - [599, 20]
898 - [899, 30]
899 - [30, 1]
901 [30, 1] [1801, 60]
902 - [901, 30]
903 - [601, 20]
904 - [451, 15]
905 - [361, 12]
906 - [301, 10]
907 - [123823410343073497682, 4111488857741309517]
908 - [102151, 3390]
909 - [80801, 2680]
910 - [181, 6]
911 - [371832584927520, 12319363142953]
912 - [151, 5]
913 - [515734243080407, 17068312251564]
914 [5593, 185] [62563299, 2069410]
915 - [121, 4]
916 - [5848201, 193230]
917 - [823604599, 27197820]
918 - [4120901, 136010]
919 - [4481603010937119451551263720, 147834442396536759781499589]
920 - [91, 3]
921 - [2522057712835735, 83104627139412]
922 [419288307, 13808525] [351605368773852499, 11579506138834350]
923 - [638, 21]
924 - [11551, 380]
925 [882, 29] [1555849, 51156]
926 - [304560297142335, 10008472361032]
927 - [227528, 7473]
928 - [768555217, 25229061]
929 [81317086468, 2667927065] [13224937103288377430049, 433896111669844912840]
930 - [61, 2]
931 - [6681448801, 218975640]
932 - [1072400673, 35127652]
933 - [75263, 2464]
934 - [3034565, 99294]
935 - [1376, 45]
936 - [5201, 170]
937 [490226695010796, 16015008052621] [480644425002415999597113107233, 15701968936415353889062192632]
938 - [17151, 560]
939 - [122695, 4004]
940 - [4231, 138]
941 [731069390, 23832181] [1068924905989944201, 34845956052079180]
942 - [106133, 3458]
943 - [737, 24]
944 - [561799, 18285]
945 - [275561, 8964]
946 - [45225786400145, 1470417148788]
947 - [13509645362, 439004487]
948 - [228151, 7410]
949 [17458843558590, 566738044393] [609622436806639069525576201, 19789181711517243032971740]
950 - [202501, 6570]
951 - [224208076, 7270445]
952 - [11663, 378]
953 [2746864744, 88979677] [15090531843660371073, 488830275367615376]
954 - [32080051, 1038630]
955 - [2095256249, 67800900]
956 - [76759023628799, 2482564242480]
957 - [14849, 480]
958 - [16762522330425599, 541572514048560]
959 - [960, 31]
960 - [31, 1]
962 [31, 1] [1923, 62]
963 - [962, 31]
964 - [10085143557001249, 324820602522300]
965 [14942, 481] [446526729, 14374204]
966 - [57499, 1850]
967 - [4649532557817485528, 149518887194649693]
968 - [19601, 630]
969 - [13588951, 436540]
970 [328173, 10537] [215395035859, 6915917802]
971 - [12479806786330, 400496058813]
972 - [9863382151, 316368130]
973 - [903223, 28956]
974 - [488825745235215, 15662987185124]
975 - [1249, 40]
976 - [1766319049, 56538495]
977 [7376748868, 236003105] [108832847723078562849, 3481871275306470280]
978 - [118337, 3784]
979 - [360449, 11520]
980 - [51841, 1656]
981 - [158070671986249, 5046808151700]
982 - [8837, 282]
983 - [284088, 9061]
984 - [88805, 2831]
985 [408, 13] [332929, 10608]
986 [157, 5] [49299, 1570]
987 - [377, 12]
988 - [14549450527, 462879684]
989 - [550271588560695, 17497618534396]
990 - [881, 28]
991 - [379516400906811930638014896080, 12055735790331359447442538767]
992 - [63, 2]
993 - [2647, 84]
994 - [1135, 36]
995 - [8835999, 280120]
996 - [8553815, 271038]
997 [84906, 2689] [14418057673, 456624468]
998 - [984076901, 31150410]
999 - [102688615, 3248924]

■1000≦D≦9999についての結果は、以下のようになる。
[2005.04.24追記]
浮動小数点計算で十分な精度がなければ、正しい結果は得られない。
2≦D≦9999に対しては、10000桁の精度をとれば、十分のようである。
再計算して、誤りを訂正した。


[参考文献]


Last Update: 2021.11.21
H.Nakao

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