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Egyptian Fractions Problem (1/x)+(1/y)+(1/z)=(4/2006)

[2006.01.02]単位分数の問題 (1/x)+(1/y)+(1/z)=(4/2006)

■単位分数の問題
正有理数(m/n)をk個の正の単位分数1/x1,1/x2,...,1/xkの和で表現する。
つまり、
    (m/n)=(1/x1)+(1/x2)+...+(1/xk)
を満たすk個の正整数の組(x1,x2,...,xk)を求める。

特に、k=3,m=4,5の場合に、以下の予想があるが、どちらも未解決である[1]。

■Erd\"os,Straus予想
全てのn > 1に対して、
    (4/n)=(1/x)+(1/y)+(1/z) ----------- (1)
は、正整数解(x,y,z)を持つ。

この予想は、n <= 1003162753(51000000番目の素数)に対して成立することが、確認されている。

■Sierpi\'nski予想
全てのn > 1に対して、
    (5/n)=(1/x)+(1/y)+(1/z) ----------- (2)
は、正整数解(x,y,z)を持つ。

この予想は、n <= 1057438801に対して成立することが、確認されている。

■n > 0を固定するとき、(1)の正整数解(x,y,z)は有限個であることを証明する。
0 < x <= y <= zと仮定する。
(1)より、
    (1/x) < (4/n) <= (3/x)
    (n/4) < x <= (3n/4)
となる。よって、xは有限個の整数ceil((n+1/2)/4), ... ,floor(3n/4)のいずれかである。

次に、xを固定すると、(1)より、
    4xyz = n(xy+yz+zx)
    (4x-n)yz-nxy-nxz = 0
    ((4x-n)y-nx)((4x-n)z-nx) = (nx)2
となる。
ここで、
    (4x-n)y-nx = nxy/z > 0,
    (4x-n)z-nx = nxz/y > 0
より、(4x-n)y-nx,(4x-n)z-nxは、正整数であり、その積が整数(nx)2であるので、その組合せは、有限個である。
つまり、整数(nx)2のある約数dに対して、
    (4x-n)y-nx = d,
    (4x-n)z-nx = (nx)2/d
となるので、
    y = (d+nx)/(4x-n),
    z = ((nx)2/d+nx)/(4x-n) = nx(nx+d)/(d*(4n-x))
を得る。ここで、y,zは整数である必要があるが、その組合せも有限個である。
よって、(1)および0 < x <= y <= zを満たす正整数の組(x,y,z)は有限個である。
x,y,zを必要に応じて入れ換えることにより、(1)の正整数解(x,y,z)が全て得られるので、(1)の正整数解は有限個である。

■同様にして、n > 0を固定するとき、(2)の正整数解(x,y,z)は有限個であることも証明できる。

■n=2006に対して、(1)の正整数解(x,y,z)で、0 < x <= y <= zを満たすものを全て求める。
pari/gpにより、プログラムを作成して、(1)の正整数解(x,y,z)で0 < x <= y <= zを満たすものを全て求めると、以下のように708個の解が得られる。
gp> read("egyfrac.gp")
time = 31 ms.
gp> find(2006)
1:[502, 503507, 253518795542] 2/1003
2:[502, 503508, 126759649524] 2/1003
3:[502, 503510, 63380076515] 2/1003
4:[502, 503523, 14913344214] 2/1003
5:[502, 503540, 7456923860] 2/1003
6:[502, 503565, 4297423710] 2/1003
7:[502, 503574, 3728713683] 2/1003
8:[502, 503624, 2148963608] 2/1003
9:[502, 503742, 1074733557] 2/1003
10:[502, 503757, 1010536542] 2/1003
11:[502, 503795, 877729430] 2/1003
12:[502, 504008, 505520024] 2/1003
13:[502, 504084, 439116468] 2/1003
14:[502, 504509, 253263518] 2/1003
15:[502, 504510, 253011765] 2/1003
16:[502, 504662, 219809987] 2/1003
17:[502, 505512, 126883512] 2/1003
18:[502, 506987, 73332662] 2/1003
19:[502, 507518, 63693509] 2/1003
20:[502, 507773, 59917214] 2/1003
21:[502, 510468, 36918084] 2/1003
22:[502, 512040, 30210360] 2/1003
23:[502, 517430, 18710795] 2/1003
24:[502, 518315, 17622710] 2/1003
25:[502, 520557, 15371742] 2/1003
26:[502, 520574, 15356933] 2/1003
27:[502, 533124, 9063108] 2/1003
28:[502, 537608, 7937624] 2/1003
29:[502, 562683, 4787574] 2/1003
30:[502, 562742, 4783307] 2/1003
31:[502, 566507, 4527542] 2/1003
32:[502, 571710, 4220565] 2/1003
33:[502, 576045, 3998430] 2/1003
34:[502, 621860, 2645540] 2/1003
35:[502, 629508, 2515524] 2/1003
36:[502, 648584, 2250968] 2/1003
37:[502, 740214, 1574523] 2/1003
38:[502, 755259, 1510518] 2/1003
39:[502, 755510, 1509515] 2/1003
40:[502, 793662, 1377237] 2/1003
41:[502, 1007012, 1007012] 2/1003
42:[503, 168170, 84843278530] 2/1003
43:[503, 168266, 293742946] 2/1003
44:[503, 168504, 84757512] 2/1003
45:[503, 169330, 24541370] 2/1003
46:[503, 171020, 10090180] 2/1003
47:[503, 178062, 3027054] 2/1003
48:[503, 252506, 503506] 2/1003
49:[504, 101103, 17036259912] 2/1003
50:[504, 101104, 6388660656] 2/1003
51:[504, 101106, 2839460904] 2/1003
52:[504, 101108, 1825403832] 2/1003
53:[504, 101112, 1064861028] 2/1003
54:[504, 101115, 811346760] 2/1003
55:[504, 101116, 751696344] 2/1003
56:[504, 101122, 521616168] 2/1003
57:[504, 101124, 473327736] 2/1003
58:[504, 101126, 433223784] 2/1003
59:[504, 101133, 334143432] 2/1003
60:[504, 101136, 304318224] 2/1003
61:[504, 101150, 214842600] 2/1003
62:[504, 101160, 177561090] 2/1003
63:[504, 101178, 135308712] 2/1003
64:[504, 101184, 125366976] 2/1003
65:[504, 101185, 123850440] 2/1003
66:[504, 101192, 114182523] 2/1003
67:[504, 101218, 88524072] 2/1003
68:[504, 101220, 87020280] 2/1003
69:[504, 101232, 78972208] 2/1003
70:[504, 101244, 72288216] 2/1003
71:[504, 101269, 61455816] 2/1003
72:[504, 101286, 55774824] 2/1003
73:[504, 101303, 51056712] 2/1003
74:[504, 101304, 50803956] 2/1003
75:[504, 101320, 47075805] 2/1003
76:[504, 101367, 38731848] 2/1003
77:[504, 101388, 35891352] 2/1003
78:[504, 101416, 32695794] 2/1003
79:[504, 101421, 32184264] 2/1003
80:[504, 101448, 29677767] 2/1003
81:[504, 101480, 27171270] 2/1003
82:[504, 101507, 25364808] 2/1003
83:[504, 101556, 22635704] 2/1003
84:[504, 101592, 20978748] 2/1003
85:[504, 101598, 20725992] 2/1003
86:[504, 101745, 16007880] 2/1003
87:[504, 101796, 14838264] 2/1003
88:[504, 101808, 14587632] 2/1003
89:[504, 101864, 13522446] 2/1003
90:[504, 101952, 12132288] 2/1003
91:[504, 102102, 10326888] 2/1003
92:[504, 102204, 9380056] 2/1003
93:[504, 102306, 8593704] 2/1003
94:[504, 102312, 8551578] 2/1003
95:[504, 102424, 7835436] 2/1003
96:[504, 102663, 6650952] 2/1003
97:[504, 102690, 6539560] 2/1003
98:[504, 102816, 6066144] 2/1003
99:[504, 102837, 5993928] 2/1003
100:[504, 102952, 5627538] 2/1003
101:[504, 102984, 5533551] 2/1003
102:[504, 103014, 5448296] 2/1003
103:[504, 103191, 4995144] 2/1003
104:[504, 103368, 4612797] 2/1003
105:[504, 103530, 4311720] 2/1003
106:[504, 103768, 3935772] 2/1003
107:[504, 104040, 3580710] 2/1003
108:[504, 104076, 3538584] 2/1003
109:[504, 104312, 3285828] 2/1003
110:[504, 104958, 2752232] 2/1003
111:[504, 105264, 2557296] 2/1003
112:[504, 105315, 2527560] 2/1003
113:[504, 105336, 2515524] 2/1003
114:[504, 105672, 2337993] 2/1003
115:[504, 105728, 2310912] 2/1003
116:[504, 106200, 2106300] 2/1003
117:[504, 106672, 1936368] 2/1003
118:[504, 107100, 1805400] 2/1003
119:[504, 107576, 1680084] 2/1003
120:[504, 107793, 1628872] 2/1003
121:[504, 108324, 1516536] 2/1003
122:[504, 108360, 1509515] 2/1003
123:[504, 109032, 1390158] 2/1003
124:[504, 109599, 1304136] 2/1003
125:[504, 110466, 1192744] 2/1003
126:[504, 111333, 1100232] 2/1003
127:[504, 111384, 1095276] 2/1003
128:[504, 111510, 1083240] 2/1003
129:[504, 112200, 1022175] 2/1003
130:[504, 112336, 1011024] 2/1003
131:[504, 113634, 916776] 2/1003
132:[504, 114597, 858568] 2/1003
133:[504, 115668, 802872] 2/1003
134:[504, 117096, 740214] 2/1003
135:[504, 117351, 730184] 2/1003
136:[504, 118728, 681037] 2/1003
137:[504, 118944, 674016] 2/1003
138:[504, 120360, 631890] 2/1003
139:[504, 120596, 625464] 2/1003
140:[504, 123760, 552240] 2/1003
141:[504, 126072, 510468] 2/1003
142:[504, 126378, 505512] 2/1003
143:[504, 126504, 503506] 2/1003
144:[504, 128384, 475776] 2/1003
145:[504, 128856, 469404] 2/1003
146:[504, 131688, 435302] 2/1003
147:[504, 133875, 413000] 2/1003
148:[504, 134520, 406980] 2/1003
149:[504, 137088, 385152] 2/1003
150:[504, 139944, 364266] 2/1003
151:[504, 140420, 361080] 2/1003
152:[504, 144432, 337008] 2/1003
153:[504, 144963, 334152] 2/1003
154:[504, 148444, 317016] 2/1003
155:[504, 148680, 315945] 2/1003
156:[504, 152082, 301608] 2/1003
157:[504, 162486, 267624] 2/1003
158:[504, 162792, 266798] 2/1003
159:[504, 163548, 264792] 2/1003
160:[504, 168504, 252756] 2/1003
161:[504, 169330, 250920] 2/1003
162:[504, 176292, 237048] 2/1003
163:[504, 188496, 218064] 2/1003
164:[504, 189567, 216648] 2/1003
165:[504, 196588, 208152] 2/1003
166:[504, 197064, 207621] 2/1003
167:[505, 72360, 7330285080] 2/1003
168:[505, 72420, 86310156] 2/1003
169:[505, 72570, 24920538] 2/1003
170:[505, 72720, 14587632] 2/1003
171:[506, 56391, 28619447538] 2/1003
172:[506, 56396, 622217068] 2/1003
173:[506, 56404, 242593604] 2/1003
174:[506, 56419, 113176514] 2/1003
175:[506, 56423, 99085426] 2/1003
176:[506, 56463, 44154066] 2/1003
177:[506, 56474, 38317609] 2/1003
178:[506, 56626, 13581623] 2/1003
179:[506, 56848, 7012976] 2/1003
180:[506, 56994, 5328939] 2/1003
181:[506, 57684, 2515524] 2/1003
182:[506, 57868, 2209196] 2/1003
183:[506, 57938, 2111791] 2/1003
184:[506, 59708, 1015036] 2/1003
185:[506, 60180, 895620] 2/1003
186:[506, 63503, 503506] 2/1003
187:[506, 64515, 447810] 2/1003
188:[506, 64900, 430100] 2/1003
189:[506, 74635, 230690] 2/1003
190:[506, 77231, 208978] 2/1003
191:[506, 77418, 207621] 2/1003
192:[506, 103191, 124338] 2/1003
193:[506, 110330, 115345] 2/1003
194:[507, 46230, 2612102870] 2/1003
195:[507, 46308, 27161004] 2/1003
196:[507, 46410, 11865490] 2/1003
197:[507, 60180, 199420] 2/1003
198:[507, 86190, 99710] 2/1003
199:[508, 39372, 8676894] 2/1003
200:[508, 60180, 112395] 2/1003
201:[509, 34102, 17357918] 2/1003
202:[509, 34612, 2042108] 2/1003
203:[510, 30091, 905438190] 2/1003
204:[510, 30092, 452734140] 2/1003
205:[510, 30093, 301832790] 2/1003
206:[510, 30094, 226382115] 2/1003
207:[510, 30095, 181111710] 2/1003
208:[510, 30096, 150931440] 2/1003
209:[510, 30099, 100630990] 2/1003
210:[510, 30100, 90570900] 2/1003
211:[510, 30102, 75480765] 2/1003
212:[510, 30105, 60390630] 2/1003
213:[510, 30107, 53289390] 2/1003
214:[510, 30108, 50330540] 2/1003
215:[510, 30110, 45300495] 2/1003
216:[510, 30115, 36246414] 2/1003
217:[510, 30120, 30210360] 2/1003
218:[510, 30124, 26659740] 2/1003
219:[510, 30126, 25180315] 2/1003
220:[510, 30135, 20150270] 2/1003
221:[510, 30140, 18138252] 2/1003
222:[510, 30141, 17783190] 2/1003
223:[510, 30149, 15375990] 2/1003
224:[510, 30150, 15120225] 2/1003
225:[510, 30158, 13344915] 2/1003
226:[510, 30165, 12102198] 2/1003
227:[510, 30175, 10681950] 2/1003
228:[510, 30180, 10090180] 2/1003
229:[510, 30190, 9084171] 2/1003
230:[510, 30192, 8906640] 2/1003
231:[510, 30208, 7703040] 2/1003
232:[510, 30240, 6066144] 2/1003
233:[510, 30243, 5947790] 2/1003
234:[510, 30260, 5356020] 2/1003
235:[510, 30267, 5145390] 2/1003
236:[510, 30270, 5060135] 2/1003
237:[510, 30294, 4468365] 2/1003
238:[510, 30315, 4054126] 2/1003
239:[510, 30326, 3866565] 2/1003
240:[510, 30345, 3580710] 2/1003
241:[510, 30379, 3162990] 2/1003
242:[510, 30385, 3099270] 2/1003
243:[510, 30390, 3048117] 2/1003
244:[510, 30396, 2988940] 2/1003
245:[510, 30430, 2693055] 2/1003
246:[510, 30444, 2587740] 2/1003
247:[510, 30515, 2160462] 2/1003
248:[510, 30540, 2042108] 2/1003
249:[510, 30600, 1805400] 2/1003
250:[510, 30621, 1735190] 2/1003
251:[510, 30668, 1596540] 2/1003
252:[510, 30680, 1564680] 2/1003
253:[510, 30702, 1509515] 2/1003
254:[510, 30798, 1308915] 2/1003
255:[510, 30855, 1213630] 2/1003
256:[510, 30940, 1095276] 2/1003
257:[510, 30957, 1074390] 2/1003
258:[510, 30975, 1053150] 2/1003
259:[510, 30990, 1036099] 2/1003
260:[510, 31093, 932790] 2/1003
261:[510, 31110, 917745] 2/1003
262:[510, 31152, 882640] 2/1003
263:[510, 31246, 813315] 2/1003
264:[510, 31270, 797385] 2/1003
265:[510, 31365, 740214] 2/1003
266:[510, 31535, 656670] 2/1003
267:[510, 31565, 643926] 2/1003
268:[510, 31620, 621860] 2/1003
269:[510, 31790, 562683] 2/1003
270:[510, 31824, 552240] 2/1003
271:[510, 31860, 541620] 2/1003
272:[510, 32096, 481440] 2/1003
273:[510, 32214, 456365] 2/1003
274:[510, 32640, 385152] 2/1003
275:[510, 32691, 378190] 2/1003
276:[510, 32745, 371110] 2/1003
277:[510, 32980, 343380] 2/1003
278:[510, 33040, 337008] 2/1003
279:[510, 33099, 330990] 2/1003
280:[510, 33150, 325975] 2/1003
281:[510, 33558, 291165] 2/1003
282:[510, 33571, 290190] 2/1003
283:[510, 33630, 285855] 2/1003
284:[510, 33915, 266798] 2/1003
285:[510, 34102, 255765] 2/1003
286:[510, 34425, 238950] 2/1003
287:[510, 34515, 234702] 2/1003
288:[510, 35105, 210630] 2/1003
289:[510, 35190, 207621] 2/1003
290:[510, 35292, 204140] 2/1003
291:[510, 35400, 200600] 2/1003
292:[510, 35870, 186735] 2/1003
293:[510, 35990, 183549] 2/1003
294:[510, 36108, 180540] 2/1003
295:[510, 37052, 160140] 2/1003
296:[510, 37315, 155406] 2/1003
297:[510, 37740, 148444] 2/1003
298:[510, 38760, 134520] 2/1003
299:[510, 38940, 132396] 2/1003
300:[510, 39117, 130390] 2/1003
301:[510, 40120, 120360] 2/1003
302:[510, 40494, 117115] 2/1003
303:[510, 40533, 116790] 2/1003
304:[510, 40710, 115345] 2/1003
305:[510, 42126, 105315] 2/1003
306:[510, 43095, 99710] 2/1003
307:[510, 43365, 98294] 2/1003
308:[510, 44014, 95115] 2/1003
309:[510, 44540, 92748] 2/1003
310:[510, 45135, 90270] 2/1003
311:[510, 45390, 89267] 2/1003
312:[510, 47141, 83190] 2/1003
313:[510, 47430, 82305] 2/1003
314:[510, 47495, 82110] 2/1003
315:[510, 47790, 81243] 2/1003
316:[510, 48144, 80240] 2/1003
317:[510, 50150, 75225] 2/1003
318:[510, 50976, 73440] 2/1003
319:[510, 51765, 71862] 2/1003
320:[510, 55165, 66198] 2/1003
321:[510, 56100, 64900] 2/1003
322:[510, 56640, 64192] 2/1003
323:[510, 58990, 61419] 2/1003
324:[510, 60180, 60180] 2/1003
325:[511, 26978, 282186026] 2/1003
326:[511, 30090, 260610] 2/1003
327:[511, 34748, 120596] 2/1003
328:[512, 24480, 23109120] 2/1003
329:[512, 24544, 6675968] 2/1003
330:[512, 26078, 392704] 2/1003
331:[512, 26112, 385152] 2/1003
332:[512, 30090, 130560] 2/1003
333:[512, 30208, 128384] 2/1003
334:[513, 22372, 677133324] 2/1003
335:[513, 22420, 10290780] 2/1003
336:[513, 22572, 2515524] 2/1003
337:[513, 22610, 2118690] 2/1003
338:[513, 30090, 87210] 2/1003
339:[513, 38114, 54162] 2/1003
340:[514, 30090, 65535] 2/1003
341:[515, 19142, 34213510] 2/1003
342:[515, 20060, 413236] 2/1003
343:[515, 30090, 52530] 2/1003
344:[516, 17850, 90570900] 2/1003
345:[516, 17974, 2515524] 2/1003
346:[516, 18054, 1552644] 2/1003
347:[516, 18060, 1509515] 2/1003
348:[516, 18275, 761100] 2/1003
349:[516, 18564, 461734] 2/1003
350:[516, 25370, 60180] 2/1003
351:[516, 30090, 43860] 2/1003
352:[516, 30444, 43129] 2/1003
353:[517, 16728, 510254184] 2/1003
354:[517, 16830, 2745270] 2/1003
355:[518, 15750, 41749875] 2/1003
356:[518, 15799, 4527542] 2/1003
357:[518, 15910, 1509515] 2/1003
358:[518, 16166, 603211] 2/1003
359:[518, 17612, 148444] 2/1003
360:[518, 19647, 79254] 2/1003
361:[519, 14878, 44767902] 2/1003
362:[519, 29410, 30090] 2/1003
363:[520, 14105, 22635704] 2/1003
364:[520, 14144, 4172480] 2/1003
365:[520, 14160, 3129360] 2/1003
366:[520, 14280, 1095276] 2/1003
367:[520, 18408, 60180] 2/1003
368:[520, 26078, 30680] 2/1003
369:[520, 26520, 30090] 2/1003
370:[522, 12770, 1671484455] 2/1003
371:[522, 12784, 11580048] 2/1003
372:[522, 12789, 8551578] 2/1003
373:[522, 12818, 3403179] 2/1003
374:[522, 19647, 36482] 2/1003
375:[522, 22185, 30090] 2/1003
376:[523, 12206, 22155326] 2/1003
377:[524, 11682, 52031628] 2/1003
378:[524, 12036, 394179] 2/1003
379:[524, 13362, 92748] 2/1003
380:[525, 11210, 20009850] 2/1003
381:[525, 11220, 7723100] 2/1003
382:[525, 17850, 30090] 2/1003
383:[527, 10370, 18966730] 2/1003
384:[527, 10384, 5472368] 2/1003
385:[527, 10540, 621860] 2/1003
386:[527, 10974, 186558] 2/1003
387:[527, 15810, 30090] 2/1003
388:[527, 16048, 29264] 2/1003
389:[528, 9996, 25949616] 2/1003
390:[528, 9999, 14587632] 2/1003
391:[528, 10030, 2647920] 2/1003
392:[528, 10032, 2515524] 2/1003
393:[528, 10384, 264792] 2/1003
394:[528, 10472, 218064] 2/1003
395:[528, 14960, 30090] 2/1003
396:[530, 9328, 46779920] 2/1003
397:[530, 9381, 1594770] 2/1003
398:[530, 13515, 30090] 2/1003
399:[531, 9028, 81495756] 2/1003
400:[531, 9030, 27171270] 2/1003
401:[531, 9036, 9063108] 2/1003
402:[531, 9044, 4802364] 2/1003
403:[531, 9054, 3027054] 2/1003
404:[531, 9078, 1606806] 2/1003
405:[531, 9086, 1390158] 2/1003
406:[531, 9108, 1015036] 2/1003
407:[531, 9180, 541620] 2/1003
408:[531, 9204, 469404] 2/1003
409:[531, 9316, 290988] 2/1003
410:[531, 9486, 186558] 2/1003
411:[531, 9558, 162486] 2/1003
412:[531, 9894, 103014] 2/1003
413:[531, 10030, 90270] 2/1003
414:[531, 10404, 68204] 2/1003
415:[531, 10620, 60180] 2/1003
416:[531, 11628, 40356] 2/1003
417:[531, 12036, 36108] 2/1003
418:[531, 12508, 32436] 2/1003
419:[531, 13806, 26078] 2/1003
420:[531, 16830, 19470] 2/1003
421:[531, 18054, 18054] 2/1003
422:[532, 8778, 2515524] 2/1003
423:[532, 9027, 282492] 2/1003
424:[532, 9044, 266798] 2/1003
425:[532, 9204, 176358] 2/1003
426:[533, 9204, 108732] 2/1003
427:[534, 9027, 94518] 2/1003
428:[534, 9078, 89267] 2/1003
429:[534, 9204, 78676] 2/1003
430:[535, 8024, 4292840] 2/1003
431:[535, 10914, 30090] 2/1003
432:[536, 7906, 537608] 2/1003
433:[536, 8024, 268804] 2/1003
434:[538, 7392, 1994413344] 2/1003
435:[539, 10472, 23128] 2/1003
436:[540, 7035, 50803956] 2/1003
437:[540, 7038, 12457260] 2/1003
438:[540, 7080, 1083240] 2/1003
439:[540, 7965, 60180] 2/1003
440:[540, 8160, 50976] 2/1003
441:[540, 9027, 31860] 2/1003
442:[540, 9180, 30090] 2/1003
443:[540, 9204, 29835] 2/1003
444:[542, 6715, 12631310] 2/1003
445:[543, 6562, 210226794] 2/1003
446:[544, 6420, 51514080] 2/1003
447:[544, 6426, 6066144] 2/1003
448:[544, 6431, 3498464] 2/1003
449:[544, 6432, 3225648] 2/1003
450:[544, 6477, 719328] 2/1003
451:[544, 6528, 385152] 2/1003
452:[544, 6608, 224672] 2/1003
453:[544, 6624, 207621] 2/1003
454:[544, 7344, 50976] 2/1003
455:[544, 8024, 32096] 2/1003
456:[544, 8160, 30090] 2/1003
457:[544, 9204, 21216] 2/1003
458:[544, 9440, 20060] 2/1003
459:[546, 6154, 49561239] 2/1003
460:[546, 6188, 1095276] 2/1003
461:[546, 6195, 912730] 2/1003
462:[546, 6426, 144963] 2/1003
463:[546, 7735, 30090] 2/1003
464:[546, 9204, 18564] 2/1003
465:[548, 9316, 16166] 2/1003
466:[550, 5700, 2515524] 2/1003
467:[550, 5723, 906950] 2/1003
468:[551, 5586, 8551578] 2/1003
469:[552, 5487, 5721112] 2/1003
470:[555, 5270, 406038] 2/1003
471:[555, 6290, 30090] 2/1003
472:[558, 4956, 7835436] 2/1003
473:[558, 5270, 82305] 2/1003
474:[558, 9027, 10974] 2/1003
475:[560, 4816, 1509515] 2/1003
476:[560, 5015, 112336] 2/1003
477:[560, 5712, 30090] 2/1003
478:[561, 4730, 14232570] 2/1003
479:[561, 4884, 148444] 2/1003
480:[561, 5100, 64900] 2/1003
481:[561, 5610, 30090] 2/1003
482:[561, 6018, 22066] 2/1003
483:[561, 7788, 12036] 2/1003
484:[561, 9204, 9724] 2/1003
485:[564, 4692, 127558] 2/1003
486:[566, 4403, 8649046] 2/1003
487:[568, 4284, 35891352] 2/1003
488:[570, 4180, 2515524] 2/1003
489:[570, 4484, 60180] 2/1003
490:[570, 4845, 30090] 2/1003
491:[572, 4069, 179573108] 2/1003
492:[572, 4114, 371228] 2/1003
493:[572, 7293, 9204] 2/1003
494:[578, 3791, 7604746] 2/1003
495:[578, 4012, 68204] 2/1003
496:[578, 4335, 30090] 2/1003
497:[582, 5723, 9894] 2/1003
498:[585, 3536, 552240] 2/1003
499:[585, 3540, 469404] 2/1003
500:[585, 3978, 30090] 2/1003
501:[590, 3344, 16770160] 2/1003
502:[590, 3345, 6710070] 2/1003
503:[590, 3349, 1975910] 2/1003
504:[590, 3350, 1680025] 2/1003
505:[590, 3360, 674016] 2/1003
506:[590, 3363, 571710] 2/1003
507:[590, 3366, 496485] 2/1003
508:[590, 3400, 200600] 2/1003
509:[590, 3422, 145435] 2/1003
510:[590, 3485, 82246] 2/1003
511:[590, 3536, 61360] 2/1003
512:[590, 3540, 60180] 2/1003
513:[590, 3825, 26550] 2/1003
514:[590, 3835, 26078] 2/1003
515:[590, 3910, 23069] 2/1003
516:[590, 4012, 20060] 2/1003
517:[590, 5015, 10030] 2/1003
518:[590, 5270, 9145] 2/1003
519:[590, 5310, 9027] 2/1003
520:[590, 5664, 8160] 2/1003
521:[595, 3192, 16007880] 2/1003
522:[595, 3230, 266798] 2/1003
523:[595, 3570, 30090] 2/1003
524:[595, 4130, 14042] 2/1003
525:[598, 4692, 9204] 2/1003
526:[600, 3060, 1805400] 2/1003
527:[600, 3400, 30090] 2/1003
528:[602, 3009, 1811418] 2/1003
529:[602, 3010, 1509515] 2/1003
530:[604, 2958, 3100332] 2/1003
531:[606, 2958, 172811] 2/1003
532:[609, 2842, 8551578] 2/1003
533:[609, 2958, 71862] 2/1003
534:[610, 3111, 30090] 2/1003
535:[612, 2778, 16718004] 2/1003
536:[612, 2788, 740214] 2/1003
537:[612, 2790, 621860] 2/1003
538:[612, 2822, 176292] 2/1003
539:[612, 2832, 144432] 2/1003
540:[612, 3009, 36108] 2/1003
541:[612, 3060, 30090] 2/1003
542:[612, 3978, 9204] 2/1003
543:[612, 4012, 9027] 2/1003
544:[612, 4248, 8024] 2/1003
545:[620, 2635, 621860] 2/1003
546:[624, 2832, 26078] 2/1003
547:[624, 3536, 9204] 2/1003
548:[627, 2508, 2515524] 2/1003
549:[628, 2669, 37052] 2/1003
550:[629, 2478, 1558662] 2/1003
551:[629, 2516, 148444] 2/1003
552:[630, 2478, 315945] 2/1003
553:[636, 2380, 656670] 2/1003
554:[640, 2360, 128384] 2/1003
555:[644, 2380, 47495] 2/1003
556:[646, 2243, 5028806] 2/1003
557:[646, 2244, 2515524] 2/1003
558:[646, 2246, 1258883] 2/1003
559:[646, 2261, 266798] 2/1003
560:[646, 2280, 134520] 2/1003
561:[646, 2301, 87438] 2/1003
562:[646, 2318, 68381] 2/1003
563:[646, 2360, 44840] 2/1003
564:[646, 2478, 23541] 2/1003
565:[646, 2603, 16166] 2/1003
566:[646, 2964, 9204] 2/1003
567:[646, 3363, 6726] 2/1003
568:[646, 3686, 5723] 2/1003
569:[646, 4484, 4484] 2/1003
570:[647, 2242, 417962] 2/1003
571:[648, 2242, 209304] 2/1003
572:[649, 2210, 1434290] 2/1003
573:[649, 2244, 132396] 2/1003
574:[649, 2618, 14042] 2/1003
575:[650, 2210, 325975] 2/1003
576:[650, 2242, 104975] 2/1003
577:[651, 2184, 22635704] 2/1003
578:[657, 2124, 878628] 2/1003
579:[660, 2090, 2515524] 2/1003
580:[660, 2091, 1596540] 2/1003
581:[660, 2244, 30090] 2/1003
582:[663, 2074, 280722] 2/1003
583:[663, 2210, 30090] 2/1003
584:[663, 2242, 25194] 2/1003
585:[663, 2652, 9204] 2/1003
586:[665, 2040, 16007880] 2/1003
587:[665, 2242, 22610] 2/1003
588:[669, 2006, 1342014] 2/1003
589:[670, 2006, 336005] 2/1003
590:[680, 1912, 2397170] 2/1003
591:[680, 1920, 385152] 2/1003
592:[680, 1938, 134520] 2/1003
593:[680, 2006, 40120] 2/1003
594:[680, 2040, 30090] 2/1003
595:[680, 2242, 12920] 2/1003
596:[680, 2360, 10030] 2/1003
597:[684, 1881, 2515524] 2/1003
598:[684, 2242, 11628] 2/1003
599:[690, 1836, 12457260] 2/1003
600:[690, 1955, 30090] 2/1003
601:[695, 1802, 4346530] 2/1003
602:[697, 1788, 73527924] 2/1003
603:[700, 1770, 2106300] 2/1003
604:[705, 1770, 94282] 2/1003
605:[708, 1720, 5175480] 2/1003
606:[708, 1722, 1151444] 2/1003
607:[708, 1734, 204612] 2/1003
608:[708, 1740, 145435] 2/1003
609:[708, 1770, 60180] 2/1003
610:[708, 1802, 37524] 2/1003
611:[708, 1836, 27081] 2/1003
612:[708, 2006, 12036] 2/1003
613:[708, 2091, 9676] 2/1003
614:[708, 2124, 9027] 2/1003
615:[708, 2380, 6195] 2/1003
616:[708, 2714, 4692] 2/1003
617:[708, 3009, 4012] 2/1003
618:[714, 1687, 1450338] 2/1003
619:[714, 1770, 35105] 2/1003
620:[714, 1785, 30090] 2/1003
621:[714, 2242, 6783] 2/1003
622:[722, 2242, 6137] 2/1003
623:[724, 1632, 17428128] 2/1003
624:[728, 1612, 22635704] 2/1003
625:[731, 1598, 4054126] 2/1003
626:[732, 1593, 6607764] 2/1003
627:[735, 1666, 30090] 2/1003
628:[753, 1506, 503506] 2/1003
629:[760, 1475, 3811400] 2/1003
630:[765, 1456, 65716560] 2/1003
631:[765, 1530, 30090] 2/1003
632:[765, 2006, 5310] 2/1003
633:[767, 1534, 26078] 2/1003
634:[767, 1768, 8024] 2/1003
635:[777, 1428, 148444] 2/1003
636:[784, 1392, 8551578] 2/1003
637:[792, 1368, 2515524] 2/1003
638:[799, 1410, 30090] 2/1003
639:[810, 1377, 30090] 2/1003
640:[815, 1304, 6539560] 2/1003
641:[816, 1309, 218064] 2/1003
642:[816, 1360, 30090] 2/1003
643:[816, 1416, 16048] 2/1003
644:[826, 1281, 367098] 2/1003
645:[826, 1298, 77231] 2/1003
646:[826, 1428, 12036] 2/1003
647:[826, 1593, 6426] 2/1003
648:[834, 1258, 15475287] 2/1003
649:[836, 1254, 2515524] 2/1003
650:[850, 1224, 1805400] 2/1003
651:[850, 1275, 30090] 2/1003
652:[852, 1224, 301608] 2/1003
653:[855, 1224, 134520] 2/1003
654:[856, 1224, 113634] 2/1003
655:[858, 1207, 61100754] 2/1003
656:[860, 1204, 1509515] 2/1003
657:[864, 1224, 50976] 2/1003
658:[867, 1190, 3580710] 2/1003
659:[868, 1190, 621860] 2/1003
660:[880, 1166, 46779920] 2/1003
661:[882, 1224, 23128] 2/1003
662:[884, 1180, 65195] 2/1003
663:[884, 1326, 9204] 2/1003
664:[885, 1158, 1935790] 2/1003
665:[885, 1180, 60180] 2/1003
666:[885, 1190, 42126] 2/1003
667:[885, 1224, 21240] 2/1003
668:[888, 1224, 19647] 2/1003
669:[910, 1118, 1509515] 2/1003
670:[915, 1110, 4527542] 2/1003
671:[918, 1107, 740214] 2/1003
672:[935, 1082, 3511090] 2/1003
673:[935, 1090, 141482] 2/1003
674:[935, 1100, 64900] 2/1003
675:[935, 1122, 30090] 2/1003
676:[935, 1180, 12980] 2/1003
677:[935, 1298, 6490] 2/1003
678:[935, 2090, 2242] 2/1003
679:[936, 1224, 9204] 2/1003
680:[944, 1070, 8585680] 2/1003
681:[944, 1071, 1011024] 2/1003
682:[944, 1072, 537608] 2/1003
683:[944, 1088, 64192] 2/1003
684:[944, 1224, 8496] 2/1003
685:[944, 1360, 5015] 2/1003
686:[944, 1534, 3536] 2/1003
687:[952, 1062, 505512] 2/1003
688:[952, 1064, 266798] 2/1003
689:[960, 1088, 30090] 2/1003
690:[966, 1043, 20623686] 2/1003
691:[969, 1938, 2242] 2/1003
692:[1003, 1004, 1007012] 2/1003
693:[1003, 1020, 60180] 2/1003
694:[1003, 1062, 18054] 2/1003
695:[1003, 1292, 4484] 2/1003
696:[1003, 2006, 2006] 2/1003
697:[1004, 1004, 503506] 2/1003
698:[1007, 1802, 2242] 2/1003
699:[1020, 1020, 30090] 2/1003
700:[1020, 1105, 9204] 2/1003
701:[1020, 1180, 6018] 2/1003
702:[1054, 1829, 2006] 2/1003
703:[1062, 1062, 9027] 2/1003
704:[1062, 1224, 4248] 2/1003
705:[1122, 1298, 3009] 2/1003
706:[1224, 1368, 2242] 2/1003
707:[1224, 1416, 2124] 2/1003
708:[1292, 1292, 2242] 2/1003
time = 14,818 ms.

■同様に、n=2006に対して、(2)の正整数解(x,y,z)で、0 < x <= y <= zを満たすものを全て求める。
pari/gpにより、プログラムを作成して、(2)の正整数解(x,y,z)で0 < x <= y <= zを満たすものを全て求めると、以下のように572個の解が得られる。
gp> find5(2006)
1:[402, 201604, 40643971212] 5/2006
2:[402, 201606, 13548124806] 5/2006
3:[402, 201612, 4516176004] 5/2006
4:[402, 201620, 2391011580] 5/2006
5:[402, 201654, 797138262] 5/2006
6:[402, 201662, 689079054] 5/2006
7:[402, 201670, 606825030] 5/2006
8:[402, 201756, 265847156] 5/2006
9:[402, 201780, 229827420] 5/2006
10:[402, 201804, 202409412] 5/2006
11:[402, 201892, 140837484] 5/2006
12:[402, 202134, 76743542] 5/2006
13:[402, 202206, 67604206] 5/2006
14:[402, 202470, 47080230] 5/2006
15:[402, 202606, 40723806] 5/2006
16:[402, 202742, 35885334] 5/2006
17:[402, 204204, 15827812] 5/2006
18:[402, 204612, 13709004] 5/2006
19:[402, 205020, 12096180] 5/2006
20:[402, 205084, 11877492] 5/2006
21:[402, 205556, 10483356] 5/2006
22:[402, 206092, 9255684] 5/2006
23:[402, 210630, 4704070] 5/2006
24:[402, 211854, 4166462] 5/2006
25:[402, 212046, 4093566] 5/2006
26:[402, 213462, 3628854] 5/2006
27:[402, 215070, 3219630] 5/2006
28:[402, 218654, 2585262] 5/2006
29:[402, 220966, 2300646] 5/2006
30:[402, 232932, 1498924] 5/2006
31:[402, 237180, 1344020] 5/2006
32:[402, 242004, 1207612] 5/2006
33:[402, 252756, 996156] 5/2006
34:[402, 259692, 901284] 5/2006
35:[402, 260780, 888420] 5/2006
36:[402, 268804, 806412] 5/2006
37:[402, 277916, 734196] 5/2006
38:[402, 355062, 466454] 5/2006
39:[402, 375870, 434830] 5/2006
40:[402, 379134, 430542] 5/2006
41:[402, 403206, 403206] 5/2006
42:[403, 89828, 2135840356] 5/2006
43:[403, 89830, 1396542095] 5/2006
44:[403, 89838, 585698841] 5/2006
45:[403, 89843, 429767446] 5/2006
46:[403, 89869, 180277214] 5/2006
47:[403, 89931, 75652278] 5/2006
48:[403, 90211, 20950358] 5/2006
49:[403, 90270, 18189405] 5/2006
50:[403, 91273, 5658926] 5/2006
51:[403, 91494, 4921839] 5/2006
52:[403, 92040, 3731160] 5/2006
53:[403, 93279, 2425254] 5/2006
54:[403, 93806, 2116153] 5/2006
55:[403, 95108, 1616836] 5/2006
56:[403, 95251, 1576598] 5/2006
57:[403, 102424, 730184] 5/2006
58:[403, 102765, 713310] 5/2006
59:[403, 109616, 497488] 5/2006
60:[403, 120683, 351286] 5/2006
61:[403, 137020, 260780] 5/2006
62:[403, 162006, 201603] 5/2006
63:[404, 57888, 5864228064] 5/2006
64:[404, 57970, 40640380] 5/2006
65:[404, 58174, 11751148] 5/2006
66:[404, 58378, 6888604] 5/2006
67:[404, 58548, 5130699] 5/2006
68:[404, 58882, 3427132] 5/2006
69:[404, 59590, 2026060] 5/2006
70:[404, 60180, 1519545] 5/2006
71:[404, 60802, 1207612] 5/2006
72:[404, 61812, 911727] 5/2006
73:[404, 65844, 479043] 5/2006
74:[404, 71508, 303909] 5/2006
75:[404, 81204, 201603] 5/2006
76:[405, 42760, 3473950680] 5/2006
77:[405, 42768, 214481520] 5/2006
78:[405, 42775, 117802350] 5/2006
79:[405, 42795, 51508062] 5/2006
80:[405, 42840, 22748040] 5/2006
81:[405, 42942, 10059795] 5/2006
82:[405, 43011, 7311870] 5/2006
83:[405, 43070, 5930739] 5/2006
84:[405, 43335, 3219630] 5/2006
85:[405, 43520, 2446848] 5/2006
86:[405, 44064, 1444320] 5/2006
87:[405, 44958, 874395] 5/2006
88:[405, 45135, 812430] 5/2006
89:[405, 47790, 406215] 5/2006
90:[405, 48144, 382320] 5/2006
91:[405, 48195, 379134] 5/2006
92:[405, 51920, 242352] 5/2006
93:[405, 54270, 201603] 5/2006
94:[405, 55080, 191160] 5/2006
95:[405, 65195, 124254] 5/2006
96:[405, 81243, 90270] 5/2006
97:[406, 33935, 6909471415] 5/2006
98:[406, 33936, 987096432] 5/2006
99:[406, 33943, 141042863] 5/2006
100:[406, 33983, 23942023] 5/2006
101:[406, 34017, 14049021] 5/2006
102:[406, 34075, 8249675] 5/2006
103:[406, 34102, 6922706] 5/2006
104:[406, 34220, 4072180] 5/2006
105:[406, 34272, 3449376] 5/2006
106:[406, 34510, 2036090] 5/2006
107:[406, 34515, 2018835] 5/2006
108:[406, 34916, 1207612] 5/2006
109:[406, 35105, 1018045] 5/2006
110:[406, 35931, 610827] 5/2006
111:[406, 36295, 521855] 5/2006
112:[406, 37961, 319957] 5/2006
113:[406, 37996, 317492] 5/2006
114:[406, 40803, 201603] 5/2006
115:[406, 42126, 174522] 5/2006
116:[406, 47908, 116348] 5/2006
117:[406, 62363, 74443] 5/2006
118:[407, 28342, 4225639] 5/2006
119:[407, 29359, 685462] 5/2006
120:[407, 39294, 99297] 5/2006
121:[408, 24073, 579485256] 5/2006
122:[408, 24074, 289754664] 5/2006
123:[408, 24075, 193177800] 5/2006
124:[408, 24076, 144889368] 5/2006
125:[408, 24078, 96600936] 5/2006
126:[408, 24080, 72456720] 5/2006
127:[408, 24081, 64408648] 5/2006
128:[408, 24084, 48312504] 5/2006
129:[408, 24088, 36240396] 5/2006
130:[408, 24089, 34110024] 5/2006
131:[408, 24090, 32216360] 5/2006
132:[408, 24096, 24168288] 5/2006
133:[408, 24104, 18132234] 5/2006
134:[408, 24106, 17067048] 5/2006
135:[408, 24108, 16120216] 5/2006
136:[408, 24120, 12096180] 5/2006
137:[408, 24123, 11386056] 5/2006
138:[408, 24131, 9845448] 5/2006
139:[408, 24136, 9078153] 5/2006
140:[408, 24140, 8545560] 5/2006
141:[408, 24144, 8072144] 5/2006
142:[408, 24168, 6060126] 5/2006
143:[408, 24174, 5705064] 5/2006
144:[408, 24190, 4934760] 5/2006
145:[408, 24208, 4284816] 5/2006
146:[408, 24216, 4048108] 5/2006
147:[408, 24225, 3811400] 5/2006
148:[408, 24249, 3297864] 5/2006
149:[408, 24264, 3042099] 5/2006
150:[408, 24276, 2864568] 5/2006
151:[408, 24308, 2479416] 5/2006
152:[408, 24344, 2154444] 5/2006
153:[408, 24360, 2036090] 5/2006
154:[408, 24361, 2029128] 5/2006
155:[408, 24378, 1917736] 5/2006
156:[408, 24426, 1660968] 5/2006
157:[408, 24480, 1444320] 5/2006
158:[408, 24544, 1251744] 5/2006
159:[408, 24603, 1115336] 5/2006
160:[408, 24616, 1089258] 5/2006
161:[408, 24648, 1030081] 5/2006
162:[408, 24650, 1026600] 5/2006
163:[408, 24684, 970904] 5/2006
164:[408, 24780, 842520] 5/2006
165:[408, 24888, 734196] 5/2006
166:[408, 24939, 692424] 5/2006
167:[408, 25016, 637908] 5/2006
168:[408, 25075, 601800] 5/2006
169:[408, 25134, 569704] 5/2006
170:[408, 25160, 556665] 5/2006
171:[408, 25228, 525336] 5/2006
172:[408, 25296, 497488] 5/2006
173:[408, 25488, 433296] 5/2006
174:[408, 25704, 379134] 5/2006
175:[408, 25806, 358248] 5/2006
176:[408, 25960, 330990] 5/2006
177:[408, 26078, 312936] 5/2006
178:[408, 26196, 296888] 5/2006
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296:[419, 9444, 3968907108] 5/2006
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322:[425, 7165, 71864950] 5/2006
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326:[425, 7225, 852550] 5/2006
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328:[425, 7990, 69325] 5/2006
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344:[432, 5712, 379134] 5/2006
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350:[438, 4779, 5930739] 5/2006
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400:[459, 4779, 9558] 5/2006
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410:[463, 3009, 2786334] 5/2006
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435:[476, 2596, 154462] 5/2006
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443:[483, 2369, 99796494] 5/2006
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447:[493, 2465, 17110] 5/2006
448:[495, 2124, 661980] 5/2006
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450:[502, 2006, 503506] 5/2006
451:[504, 1967, 142048872] 5/2006
452:[504, 2142, 24072] 5/2006
453:[510, 1881, 9433215] 5/2006
454:[510, 1887, 556665] 5/2006
455:[510, 1888, 481440] 5/2006
456:[510, 1890, 379134] 5/2006
457:[510, 1947, 55165] 5/2006
458:[510, 1989, 34515] 5/2006
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465:[516, 1806, 1207612] 5/2006
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468:[520, 1768, 260780] 5/2006
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470:[528, 1904, 13629] 5/2006
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484:[540, 3060, 3186] 5/2006
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491:[558, 1428, 7835436] 5/2006
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501:[590, 1260, 252756] 5/2006
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509:[600, 1275, 24072] 5/2006
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512:[608, 1180, 3049120] 5/2006
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515:[612, 1224, 24072] 5/2006
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518:[629, 1110, 556665] 5/2006
519:[630, 1105, 4107285] 5/2006
520:[642, 1070, 3219630] 5/2006
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524:[646, 2006, 2242] 5/2006
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527:[658, 1028, 339226636] 5/2006
528:[663, 1020, 260780] 5/2006
529:[663, 1534, 3009] 5/2006
530:[669, 1003, 1342014] 5/2006
531:[670, 1003, 336005] 5/2006
532:[670, 1005, 201603] 5/2006
533:[680, 1003, 40120] 5/2006
534:[680, 1020, 24072] 5/2006
535:[696, 986, 24072] 5/2006
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537:[697, 984, 24072] 5/2006
538:[700, 940, 8249675] 5/2006
539:[702, 1326, 3186] 5/2006
540:[708, 926, 5572668] 5/2006
541:[708, 944, 48144] 5/2006
542:[708, 1003, 12036] 5/2006
543:[708, 1020, 10030] 5/2006
544:[708, 1326, 3068] 5/2006
545:[714, 918, 379134] 5/2006
546:[714, 952, 24072] 5/2006
547:[734, 885, 5521515] 5/2006
548:[735, 885, 491470] 5/2006
549:[748, 866, 1124068] 5/2006
550:[748, 868, 281666] 5/2006
551:[748, 869, 205084] 5/2006
552:[748, 880, 51920] 5/2006
553:[748, 885, 38940] 5/2006
554:[748, 924, 13629] 5/2006
555:[748, 944, 10384] 5/2006
556:[748, 946, 10148] 5/2006
557:[748, 1180, 3245] 5/2006
558:[748, 1188, 3186] 5/2006
559:[748, 1298, 2596] 5/2006
560:[749, 864, 649077408] 5/2006
561:[760, 850, 3811400] 5/2006
562:[765, 885, 18054] 5/2006
563:[765, 1003, 5310] 5/2006
564:[767, 1326, 2301] 5/2006
565:[774, 833, 19019889] 5/2006
566:[782, 826, 322966] 5/2006
567:[804, 804, 201603] 5/2006
568:[816, 816, 24072] 5/2006
569:[826, 826, 14042] 5/2006
570:[918, 918, 3186] 5/2006
571:[1003, 1003, 2006] 5/2006
572:[1003, 1054, 1829] 5/2006
time = 11,214 ms.


[参考文献]

Last Update: 2006.01.03
H.Nakao

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