èŠçŽ nåãã°ã«ãŒãkåã«åå²ããæ¹æ³ã«ã¯ã
èŠçŽ (n-1)åãã°ã«ãŒã(k-1)åã«åå²ããnçªç®ã®èŠçŽ ãkçªç®ã®ã°ã«ãŒããšããŠè¿œå ããå Žåãšã
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èŠçŽ nåãã°ã«ãŒãkåã«åå²ããæ¹æ³ã®æ°ãS(n,k)ãšãããšã
S(n+1,k)=k*S(n,k)+S(n,k-1)
ãšããæŒžååŒã§è¡šãããšãã§ããŠãS(n,k)ã¯ã第2çš®ã¹ã¿ãŒãªã³ã°æ°ãšããŠç¥ãããŠããŸãã
第2çš®ã¹ã¿ãŒãªã³ã°æ°ã¯ãS(n,1)=1(nâ§1),S(n,n)=1(nâ§0),S(n,0)=0(nâ§1)ã§ã以äž
S(3,2)=3,S(4,2)=7,S(5,2)=15,S(6,2)=31,...
S(4,3)=6,S(5,3)=25,S(6,3)=90,...
S(5,4)=10,S(6,4)=65,...
S(6,5)=15,...
...
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æºæ°éŠã®éŠå³ã®ç·æ°ã¯5åã®èŠçŽ ã5å以äžã®ã°ã«ãŒãã«åå²ããæ¹æ³ã®æ°ãªã®ã§ã
S(5,1)+S(5,2)+S(5,3)+S(5,4)+S(5,5)=52ãšãªããŸãã
éŠæšã®çš®é¡ã3çš®é¡ã ãšãéŠå³ã®ç·æ°ã¯
S(3,1)+S(3,2)+S(3,3)=5ã§
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S(4,1)+S(4,2)+S(4,3)+S(4,4)=15ã§
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S(6,1)+S(6,2)+S(6,3)+S(6,4)+S(6,5)+S(6,6)=203ãšãªããŸãã
ãªãããã®éŠå³ã®ç·æ°ã¯ãã«æ°ãšãã£ãŠã
https://oeis.org/A000110
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2ãåºãšããäºé²å¯Ÿæ°log_{2}(x)ãlb(x)ã§è¡šããšãlb(3),lb(5),lb(7),lb(11),lb(13)ã®é£åæ°å±éã¯ã
lb(3)=[1;1,1,2,2,3,1,5,2,23,...]
lb(5)=[2;3,9,2,2,4,6,2,1,1,3,1,18..]
lb(7)=[2;1,4,5,4,5,4,1,29,...]
lb(11)=[3;2,5,1,1,1,25,1,1,...]
lb(13)=[3;1,2,2,1,22,23,1,9,149,...]
ã§ãlb(13)ã[3;1,2,2,1,22,23,1,9]ã§è¿äŒŒãããšã
[3;1,2,2,1,22,23,1,9]-lb(13)=201130/54353-lb(13)
=-2.25897731584196266637728075571*10^-12
ãšãªã£ãŠã201130/54353ãlb(13)ã®è¿äŒŒå€ãšãããšèª€å·®ãå°ããã®ã§ãlb(3),lb(5),lb(7),lb(11)ã
54353åãããšã
54353*lb(3)=86147.4668016970019305550728204
54353*lb(5)=126203.757741412805693795471951
54353*lb(7)=152588.162078596956051793358299
54353*lb(11)=188030.486767793017766203979679
ãšãªã£ãŠãå°æ°éšåã«çç®ãããšã
0.4668016970019305550728204=[0;2,7,33,...]
0.757741412805693795471951=[0;1,3,7,...]
0.162078596956051793358299=[0;6,5,1,...]
0.486767793017766203979679=[0;2,18,2,...]
ãªã®ã§ããããããã1/2=[0;2],3/4=[0;1,3],1/6={0;6],1/2=[0;2]ã§è¿äŒŒãããšã
(86147+1/2)/54353-lb(3)=6.10790627896702020627740913203*10^-7
(126203+3/4)/54353-lb(5)=-1.42428436437616975547826382828*10^-7
(152588+1/6)/54353-lb(7)=8.44124466103963591405650940218*10^-8
(188030+1/2)/54353-lb(11)=2.43449432087167148461376935139*10^-7
ãšãªããŸããã(86147+1/2)/54353,(126203+3/4)/54353,(152588+1/6)/54353,(188030+1/2)/54353ãé£åæ°å±éãããšã
(86147+1/2)/54353=[1;1,1,2,2,3,1,5,2,2,1,2,1,12]
(126203+3/4)/54353=[2;3,9,2,2,4,4,1,1,6,1,1,2]
(152588+1/6)/54353=[2;1,4,5,4,5,5,1,4,1,1,2,3]
(188030+1/2)/54353=[3;2,5,1,1,1,25,1,4,1,1,4,2]
ããã
lb(3)ã§[1;1,1,2,2,3,1,5,2,...]ã
lb(5)ã§[2;3,9,2,2,4,...]ã
lb(7)ã§[2;1,4,5,4,5,...]ã
lb(11)ã§[3;2,5,1,1,1,25,1,...]
ãŸã§äžèŽããŠããŠãlb(3),lb(5),lb(7),lb(11),lb(13)ã忝ã54353*12=652236ã®åæ°ã§æ¯èŒçããè¿äŒŒã§ãããã§ãã
lb(3),lb(5),lb(7),lb(11),lb(13)ãéåžžã«é£åæ°ããã®æã¡åãåæ°ã§
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lb(3)=301994/190537
lb(5)=227268/97879ã(6æ¡ã§ãããã®ã¯åããªãã£ãã)
lb(7)=1273419/453601
lb(11)=1444074/417431
lb(13)=201130/54353
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kuiperbeltæ°ã«ããæ§æã§ã¯
lb(3)=172295/108706=1033770/652236
lb(5)=504815/217412=1514445/652236
lb(7)=915529/326118=1831058/652236
lb(11)=376061/108706=2256366/652236
lb(13)=201130/54353=2413560/652236
ããã§åãã®æ¹ã®ç°ãªãåæ°ã652236ãžãšçµ±äžãããšããã°ããããã®ååã¯
lb(3)-->round(652236/190537*301994)=1033770
lb(5)-->round(652236/97879*227268)=1514445
lb(7)-->round(652236/453601*1273419)=1831058
lb(11)-->round(652236/417431*1444074)=2256366
lb(13)-->round(652236/54353*201130)=2413560
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16=42-15*sqrt(3)=>0.019237886466840597088304877411914495791
24=123-70*sqrt(2)=>0.0050506338833465838817893053211345001350
32=276-63*sqrt(15)=>0.0020491889327362337062798137088245173490
40=525-198*sqrt(6)=>0.0010309289307365569377532082335043901511
48=894-143*sqrt(35)=>0.00059101675490591287205430668876207606729
56=1407-780*sqrt(3)=>0.00037009627571104859185362541955378301120
64=2088-765*sqrt(7)=>0.00024703558819826626394846596577432967552
72=2961-1292*sqrt(5)=>0.00017307027171223934761999919110380885978
80=4050-1197*sqrt(11)=>0.00012594458638060942551420518803934464147
88=5379-906*sqrt(30)=>9.4500095344005671898144251386011596633 E-5
96=6972-575*sqrt(143)=>7.2716696137850341565187520446959431717 E-5
104=8853-1350*sqrt(42)=>5.7149388688195943961281204512114323297 E-5
112=11046-783*sqrt(195)=>4.5728919063980887084329367575848946092 E-5
120=13575-899*sqrt(224)=>3.7160906777440839558589688710914293688 E-5
128=16464-1023*sqrt(255)=>3.0607247824951139183948832040071781991 E-5
136=19737-13860*sqrt(2)=>2.5508902623608594282453584631070366255 E-5
144=23418-1295*sqrt(323)=>2.1483200147407576879212274060209928031 E-5
152=27531-8658*sqrt(10)=>1.8262171743553579692301522935038486629 E-5
160=32100-1599*sqrt(399)=>1.5654351913667168657434798770696208115 E-5
168=37149-3526*sqrt(110)=>1.3520456453081349107645400277878002460 E-5
176=42702-1935*sqrt(483)=>1.1757513052464834609167108976471561999 E-5
184=48783-8460*sqrt(33)=>1.0288277537663826978867857967869962110 E-5
192=55416-11515*sqrt(23)=>9.0540344785055006911665795110653250388 E-6
200=62625-9996*sqrt(39)=>8.0096115343544563630858145670082038073 E-6
208=70434-40545*sqrt(3)=>7.1198701339296880836444048825359176973 E-6
216=78867-5830*sqrt(182)=>6.3571982558417182849809728108097647648 E-6
224=87948-9405*sqrt(87)=>5.6996944965601575244751615724698325284 E-6
232=97701-6726*sqrt(210)=>5.1298361531682648788761153900878672073 E-6
240=108150-3599*sqrt(899)=>4.6334908721224596314615983300244219419 E-6
248=119319-30744*sqrt(15)=>4.1991752820486645490899063644723485097 E-6
256=131232-4095*sqrt(1023)=>3.8174932812674583700545221491687345189 E-6
264=143913-34840*sqrt(17)=>3.4807064442220806178631564440052527937 E-6
272=157386-4623*sqrt(1155)=>3.1824025866890528095006338533945508999 E-6
280=171675-29394*sqrt(34)=>2.9172379591251503174251310489710550738 E-6
288=186804-5183*sqrt(1295)=>2.6807351648308626097063006717446215453 E-6
296=202797-32850*sqrt(38)=>2.4691236092811802692280772010616819375 E-6
304=219678-5775*sqrt(1443)=>2.2792126687875382004922224433354285596 E-6
312=237471-24332*sqrt(95)=>2.1082902188077300200429255193613423229 E-6
320=256200-6399*sqrt(1599)=>1.9540409567059134673684620949682723996 E-6
328=275889-26892*sqrt(105)=>1.8144802784198278034278698655600337535 E-6
336=296562-7055*sqrt(1763)=>1.6879004543876111616534704993720122179 E-6
344=318243-14790*sqrt(462)=>1.5728265895810835886254647889304459244 E-6
352=340956-23229*sqrt(215)=>1.4679804112725554698240184211417630657 E-6
360=364725-16198*sqrt(506)=>1.3722503533570500336644416285396892947 E-6
368=389574-25389*sqrt(235)=>1.2846667317585679159772552841492182749 E-6
376=415527-35340*sqrt(138)=>1.2043810565329850268087220171954593041 E-6
384=442608-64505*sqrt(47)=>1.1306487210116121767818178054760610574 E-6
392=470841-192060*sqrt(6)=>1.0628144602296206119864992601618119558 E-6
400=500250-69993*sqrt(51)=>1.0003000900280090029710013420615528427 E-6
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(2*n+1)*sqrt(n+1)â(2*n-1)*sqrt(n-1)+4*sqrt(n)
(2*n-1)*sqrt(n-1)â(2*n+1)*sqrt(n+1)-4*sqrt(n)
4*sqrt(n)â(2*n+1)*sqrt(n+1)-(2*n-1)*sqrt(n-1)
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gp > forprime(p=2,100,for(n=1,1000000,if(frac(n*sqrt(p)) < 0.000001,print(n";"p))))
978122;3
902702;7
283009;17
566018;17
345777;19
254813;29
509626;29
528641;41
424802;43
829254;47
528896;53
951113;61
977001;89
594030;97
ã
gp > forprime(p=2,100,for(n=1,10000000,if(frac(n*sqrt(p)) > 0.9999999,print(n";"p))))
9369319;2
7865521;3
7465176;5
3096720;7
9504180;11
2298912;17
4508361;19
9016722;19
5412001;23
8193638;29
9600319;31
1311360;41
3697884;43
7395768;43
6191808;47
8142716;61
2874480;71
5748960;71
8623440;71
4684249;73
4121279;79
8242558;79
9005009;89
6377352;97
ãªãçŽ æ°ãæŽæ°ãçµã¿åãããã°(bãšpãäžã®çµã¿åããã«åã£ãŠããæå³)
a+b*sqrt(p)
ã¯aã調ç¯ããããšã§äžåã®ç¡çæ°ã§ãªããŒã§ãèªç¶æ°ãã¹ãŠã奜ããªè¿äŒŒç²ŸåºŠã§äœãåºãããšã¯å¯èœãšãªããã§ããã
ããã(*)ã®çåŒãäœãåºããã»ã³ã¹ã«ã»ã©ã»ã©æå¿ããŸããã
ãªãã»ã©ããã¯ã8ã®åæ°ã«ã ã䜿ããåŒããã£ãã®ã§ãããããããããŸããã
GAI ãããäœããããã®ãå
šãããããªããã§ããã
16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒã§ãããã®ããã«æ±ãããŠããçç±ã¯ã©ãã«ãããã§ãïŒ
> 16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒã§ãããã®ããã«æ±ãããŠããçç±ã¯ã©ãã«ãããã§ãïŒ
ãããããããžã®è¿äºã®äžã®
8*nâ4*n^3+5*n-(4*n^2-1)*sqrt(n^2-1)â€
ãªãåŒãã
n=2,7,26ã§çãŸããåŒã以äžã®åŒãšãªãæå³ã§ãã
16=42-15*sqrt(3)
56=1407-780*sqrt(3)
208=70434-40545*sqrt(3)
ãã®
16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒãšããã€ããã¯ãããŸããã
äžã®3ã€ã®åŒãã
sqrt(3)=(42-16)/15=26/15
sqrt(3)=(1407-56)/780=1351/780
sqrt(3)=(70434-208)/40545=70226/40545
çãçºçããŸãã(åã«æ²ç€ºããŠããäžããéžãã§ããã ãã§ãã)
ããã«sqrt(3)ã®é£åæ°ãéäžã§æã¡åã£ãŠåæ°ãšããŠããæé ãèªåã§ããããŠãããš
gp > contfracpnqn(contfrac(sqrt(3)),20)
%208 =
[1 2 5 7 19 26 71 97 265 362 989 1351 3691 5042 13775 18817 51409 70226 191861 262087 716035]
[1 1 3 4 11 15 41 56 153 209 571 780 2131 2911 7953 10864 29681 40545 110771 151316 413403]
ã®æ§ã«å³ã«è¡ãã«ãããã£ãŠããsqrt(3)ãžè¿äŒŒããŠãããŸãã
ãã®æµãã®äžã«
26/15,1351/780,70226/40545ãããŸããã16=42-15*sqrt(3)ãç¹å¥ã®åŒãšã®è§£é㯠"ã¯ãŠïŒ"
ãšæã£ãŠããŸããŸãã
ãããå
·äœçãªæ¹ãããããšæã£ãŠ1ã€æœåºããã質åã®æå³ãäŒãããŸããã§ããã
5çªã®åŒã§ã8n=ãã®åœ¢ã«ããåŒã ãç¹å¥èŠããŠããã®ã¯ãªãã§ããïŒ
ãã®åŒãäœããšãã«ããã®åã«èšèŒãããåŒãå¹³æ¹ããŠæŽçããŠåŸããã
4*n^3-3*n-(4*n^2-1)*sqrt(n^2-1)â0
ã®äž¡èŸºã«çªç¶ 8n ãå ããŠã§ããã®ã 5 çªã§ãããã
ããã§äž¡èŸºã« 7n ãè¶³ãã° 7 ã®åæ°ã®åŒãã§ããã§ããããã9n ãè¶³ãã° 9 ã®åæ°ã®åŒãã§ããããããªãã§ããããïŒ
8n ãéžæããå¿
ç¶æ§ã¯ãªãã§ãããïŒ
> 5çªã®åŒã§ã8n=ãã®åœ¢ã«ããåŒã ãç¹å¥èŠããŠããã®ã¯ãªãã§ããïŒ
ãã®åŒãäœããšãã«ããã®åã«èšèŒãããåŒãå¹³æ¹ããŠæŽçããŠåŸããã
4*n^3-3*n-(4*n^2-1)*sqrt(n^2-1)â0
ãšèšèŒãããŠããéšåã¯
8*nâ4*n^3+5*n-(4*n^2-1)*sqrt(n^2-1)
ããªã
3*nâ4*n^3-(4*n^2-1)*sqrt(n^2-1)
ãšããªãã®ãïŒ
ãšè§£éããŠããã§ããïŒ
çç±ã¯ããããããšãæ°ä»ããªãã£ãã§ãã
2ã§å²ããããããã§çµãããšæã蟌ã¿ããã®ãŸãŸã§æ°å€ã§ç¢ºèªã«è¡ã£ãŠããŸããã
ããããã¯
6â32-15*sqrt(3)
9â108-35*sqrt(8)=108-70*sqrt(2)
12â256-63*sqrt(15)

ã§æ²èŒããŠããããšã§ãããã
ãç²æ«æ§ã§ããã
> 3*nâ4*n^3-(4*n^2-1)*sqrt(n^2-1)
ããã«èšãã°ã3n ãš 4n^3 ãå·Šå³ã«åããæå³ããªããšæããŸãã
n ã¯å
·äœçãªæŽæ°ã代å
¥ããããšãæ³å®ããŠãããã§ããããããæŽæ°ã®è¿äŒŒåŒã«æŽæ°ãè¶³ãé
ããã£ãŠã¯æçŸ©ãèãã§ãã
äžæ¹ã§ããããéã«å¹³æ¹æ ¹ã®æçè¿äŒŒåŒãšããŠ
â(n^2-1) = (4n^3-3n)/(4n^2-1)
ãšããåŒã¯äœãã«äœ¿ãããã§ããã
DD++ããããã®ã¢ããã€ã¹ãåããŠsqrt(n^2-1)â(4*n^3-3*n)/(4*n^2-1)
ã®æŽ»çšãèŠãŠã¿ãŸããã
æéã®é¢ä¿ã§ããã°ã©ã ã®ãŸãŸã®å§¿ã§ç³ãèš³ãããŸããã
gp > for(n=1,100,if(core(n^2-1)==3,\
print(n,";sqrt(3)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/3))))
2;sqrt(3)=26/15
7;sqrt(3)=1351/780
26;sqrt(3)=70226/40545
97;sqrt(3)=3650401/2107560
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(3)),24)
%227 =
[1 2 5 7 19 26 71 97 265 362 989 1351
3691 5042 13775 18817 51409 70226
191861 262087 716035 978122 2672279 3650401
9973081]
[1 1 3 4 11 15 41 56 153 209 571 780
2131 2911 7953 10864 29681 40545
110771 151316 413403 564719 1542841 2107560
5757961]
-----------------------------------------------------------
for(n=1,10000,if(core(n^2-1)==5,\
print(n,";sqrt(5)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/5))))
9;sqrt(5)=2889/1292
161;sqrt(5)=16692641/7465176
2889;sqrt(5)=96450076809/43133785636
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(5)),20)
%222 =
[2 9 38 161 682 2889 12238 51841 219602 930249 3940598 16692641
70711162 299537289 1268860318 5374978561 22768774562 96450076809
408569081798 1730726404001 7331474697802]
[1 4 17 72 305 1292 5473 23184 98209 416020 1762289 7465176
31622993 133957148 567451585 2403763488 10182505537 43133785636
182717648081 774004377960 3278735159921]
-------------------------------------------------------------
gp > for(n=1,10000,if(core(n^2-1)==6,\
print(n,";sqrt(6)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/6))))
5;sqrt(6)=485/198
49;sqrt(6)=470449/192060
485;sqrt(6)=456335045/186298002
4801;sqrt(6)=442644523201/180708869880
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(6)),24)
%226 =
[2 5 22 49 218 485 2158 4801 21362 47525 211462 470449
2093258 4656965 20721118 46099201 205117922 456335045
2030458102 4517251249 20099463098 44716177445 198964172878
442644523201 1969542265682]
[1 2 9 20 89 198 881 1960 8721 19402 86329 192060
854569 1901198 8459361 18819920 83739041 186298002
828931049 1844160100 8205571449 18255302998 81226783441
180708869880 804062262961]
-------------------------------------------------------------
gp > for(n=1,10000,if(core(n^2-1)==7,\
print(n,";sqrt(7)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/7))))
8;sqrt(7)=2024/765
127;sqrt(7)=8193151/3096720
2024;sqrt(7)=33165873224/12535521795
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(7)),36)
%233 =
[2 3 5 8 37 45 82 127 590 717 1307 2024
9403 11427 20830 32257 149858 182115 331973
514088 2388325 2902413 5290738 8193151
38063342 46256493 84319835 130576328 606625147 737201475
1343826622 2081028097 9667939010 11748967107 21416906117 33165873224
154080399013]
[1 1 2 3 14 17 31 48 223 271 494 765
3554 4319 7873 12192 56641 68833 125474
194307 902702 1097009 1999711 3096720
14386591 17483311 31869902 49353213 229282754 278635967
507918721 786554688 3654137473 4440692161 8094829634 12535521795
58236916814]
è¿äŒŒã¹ããŒãã皌ããŸãã
次ã®èªç¶æ°nã¯2ã€ã®ç¡çæ°ãçµåãããšæ®ãã©äžèŽããå€ã«è¿ã¥ããã
nâ ç¡çæ°ã§ã®è¡šç€ºã ââ2æ°ã®èª€å·®ã®çµ¶å¯Ÿå€
3â 5*sqrt(3)-4*sqrt(2) ââ0.0033997883520062724304768106905695204447
8â 9*sqrt(5)-7*sqrt(3) ââ0.00025614451596621299043862804037955033911
12â 19*sqrt(10)-34*sqrt(2) ââ1.4422513975648721560721091917469468432 E-5
14â 5*sqrt(2)+4*sqrt(3) ââ0.00072895785901558188177101292802013936790
16â 33*sqrt(17)-31*sqrt(15) ââ1.9129528747035492978538901538940856089 E-6
18â 11*sqrt(6)-4*sqrt(5) ââ0.00011526061580029453343014683970036989732
20â 51*sqrt(26)-98*sqrt(6) ââ4.0048057270810559023898452505417707900 E-7
24â 73*sqrt(37)-71*sqrt(35) ââ1.1169729827469664119887608509554651446 E-7
28â 495*sqrt(2)-388*sqrt(3) ââ3.7957659268186737979522070518583296012 E-8
32â 129*sqrt(65)-381*sqrt(7) ââ1.4903890174177504572628253724468858932 E-8
36â 163*sqrt(82)-644*sqrt(5) ââ6.5343456440228885554609297031481511666 E-9
40â 201*sqrt(101)-597*sqrt(11)ââ3.1252343924574036722748632756462662144 E-9
51â 15*sqrt(7)+8*sqrt(2) ââ2.1835046380752062253901733508985792273 E-5
57â 21*sqrt(11)-4*sqrt(10) ââ9.9567898795034180132923535462276858634 E-6
124â 29*sqrt(14)+4*sqrt(15) ââ2.3987260322773542251636904718057150093 E-6
132â 105*sqrt(2)-4*sqrt(17) ââ1.5467043379248916766181219901488696439 E-6
245â 47*sqrt(23)+8*sqrt(6) ââ4.6203675675934213838670625462524820104 E-7
255â 159*sqrt(3)-4*sqrt(26) ââ3.4908035035075107186334257838554981487 E-7
426â 69*sqrt(34)+4*sqrt(35) ââ1.2327580333941413828310829952623491724 E-7
438â 75*sqrt(38)-4*sqrt(37) ââ1.0148035501276569162826004864350052660 E-7
679â 95*sqrt(47)+16*sqrt(3) ââ4.0798771434653070872845390882949520266 E-8
693â 101*sqrt(51)-20*sqrt(2) ââ3.5365948821905606463899831583875575438 E-8
1016â 125*sqrt(62)+4*sqrt(63) ââ1.5748535453351304854827891931151138080 E-8
1032â 131*sqrt(66)-4*sqrt(65) ââ1.4116608610778294521636201128152863637 E-8
1449â 159*sqrt(79)+16*sqrt(5) ââ6.8247376929139949103118573138778558128 E-9
1467â 165*sqrt(83)-4*sqrt(82) ââ6.2596490147330746056941507128262552391 E-9
1990â 1379*sqrt(2)+12*sqrt(11) ââ3.2371295130920564747781094499907966819 E-9
ãªã2024â64778373-1536210*sqrt(1778)(â2024.0000000077188666499249599539865432)
1825346970â1881803*sqrt(940902)-4*sqrt(940901)(â1825346970.0000000000000000000000386764)
ãªãç¡çæ°ã§ããªãè¿ã¥ãã(äžã®æ°ã¯0ïœ9ã®æ°åãäžåºŠé¡ãåºãã¿ã€ãã®æ°)
ããããã°ã©ã ãPARI/GPã§èµ°ãããŠãããšãããã©ãããŠãããã°ã©ã ãç¹å®ã®å€ã§ã¯
çµæããããåŸ
ã£ãŠãçµäºããããã®åå ãäžã€ãã€æœ°ããŠãããšãããªããšæã£ãŠãããªã
次ã®ãããªèšç®ãè¡ãããŠããããšã倿ããŸããã
ãã®æ§ãªããšã«ãªã£ãŠããŸãã®ã¯ãç§ã䜿ã£ãŠãããœããã«éãã®ã§ããããïŒ
çããã䜿ãããŠãããœããã§ã¯åŠäœãªãçµæãè¿ããŠããããæããŠæ¬²ããã
gp > for(n=1,20,print(n";"floor(log(10^n)/log(10))))
1;1
2;2
3;2
4;4
5;5
6;5
7;7
8;8
9;9
10;10
11;10
12;11
13;12
14;14
15;14
16;16
17;16
18;18
19;19
20;20
3,6,11,12,13,15,17ã§ãã¡ãã®ææãè£åãããŠããŸããŸããã
æå
ã§ã¯åæ§ã§ããã
PARI ã¯ãã²ãšãã³å°æ°ãæ±ãããšã«ãªããšïŒé²æ°ã§å
éšè¡šçŸããã®ããªïŒããšæããŸãããããã£ãšã¿ãããã ãã§ã¯ããŸã説æã§ããªããããªïŒ
? for(n=1,20,print(n";"floor(log(10^n)/log(10))))
1;1
2;2
3;2
4;4
5;5
6;5
7;7
8;8
9;9
10;10
11;10
12;11
13;12
14;14
15;14
16;16
17;16
18;18
19;19
20;20
JavaScript ã§ã¯ä»¥äžã®éãã§ãã
for (let n = 1; n <= 20; n++) {
console.log(n + ";" + Math.floor(Math.log(Math.pow(10, n)) / Math.log(10)));
}
äžã RUN ãããš
"1;1"
"2;2"
"3;2"
"4;4"
"5;5"
"6;5"
"7;7"
"8;8"
"9;8"
"10;10"
"11;11"
"12;11"
"13;12"
"14;14"
"15;14"
"16;16"
"17;17"
"18;17"
"19;19"
"20;20"
ãšãªããŸãã
JavaScript ã§ã¯ãã¶ãå°æ°ç¹ä»¥äžã¯ãæå¯ãã®ïŒé²æ°ã§æããŠããã®ã§âŠâŠ
ä»ã®èšç®ãœããã§ã調æ»ããŠã¿ãã
ïŒsageMathã®ãœãã
sage: for i in range(21) :print(i,floor(ln(10^i)/ln(10)));
(0, 0)
(1, 1)
(2, 2)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
(7, 7)
(8, 8)
(9, 9)
(10, 10)
(11, 11)
(12, 12)
(13, 13)
(14, 14)
(15, 15)
(16, 16)
(17, 17)
(18, 18)
(19, 19)
(20, 20)
å
šéšäžæãèµ°ã
ïŒRubyã®ãœãã
irb(main):001:0> include Math
=> Object
irb(main):012:0> 0.upto(20){|i| print i,";",log(10**i)/log(10),"\n"}
0;0.0
1;1.0
2;2.0
3;2.9999999999999996
4;4.0
5;5.0
6;5.999999999999999
7;7.0
8;8.0
9;8.999999999999998
10;10.0
11;11.0
12;11.999999999999998
13;12.999999999999998
14;14.0
15;14.999999999999998
16;16.0
17;17.0
18;17.999999999999996
19;19.0
20;20.0
3,6,9,12,13,15,18ã§äžæããããªããªãã
ïŒMaximaã®ãœãã
(%i13) for i :1 thru 20 do
print(float(log(10^i)/log(10)));
1.0" "
2.0" "
3.0" "
4.0" "
5.0" "
6.0" "
7.0" "
8.0" "
8.999999999999998" "
9.999999999999998" "
11.0" "
12.0" "
13.0" "
14.0" "
15.0" "
16.0" "
17.0" "
18.0" "
19.0" "
20.0" "
9,10ã§é£ç¹
PARI ã§ãåºé¢æ°ã䜿ãåã«åŸ®éãªäžé§ãã¯ãããŸããã 258,259ã§ç Žç¶»ã
for(n=257,260,print(n";"floor(10^(-36)+log(10^n)/log(10))))
257;257
258;257
259;258
260;260
PARI ã«ãŠã
n = 308 ãŸã§ã®ç¯å²ã§åŸ®å°éãè¶³ããã¹ããããŸããã
(javascriptã ãš10^308ãè¶
ãããšéäžèšç®çµæã«ç¡é倧ãçŸããæ±ãã«ãªã£ãã®ã§âŠâŠPARIã§ã¯ã©ããªã®ãããããããšããããã§ã)
埮å°éãšããŠã¯ã2 ^{-119}ãš2 ^{- 120} ãšã®ããã ã«å氎嶺ããããŸãã以äžã
? i = -120; for(n = 1, 308, if(n == floor(2^i + log(10^n)/log(10)), next, print(n, ";", floor(2^i + log(10^n)/log(10))) ) ); print("END")
ã
äžãèµ°ããããš
258;257
259;258
265;264
266;265
271;270
272;271
277;276
278;277
283;282
284;283
290;289
291;290
296;295
297;296
302;301
303;302
308;307
END
ã
ãšãªã
? i = -119; for(n = 1, 308, if(n == floor(2^i + log(10^n)/log(10)), next, print(n, ";", floor(2^i + log(10^n)/log(10))) ) ); print("END")
äžãèµ°ããããš
ã
END
ãšãªããŸãã
ããããããã®ãä»äºã§
ããã ãã®çŽ æ°ã«å¯Ÿããåžžçšå¯Ÿæ°å€ãå
±é忝ã§
log2 = 360565/1197771 = 0.3010299966 (çå€ 0.3010299957)
log3 = 571482/1197771 = 0.4771212527 (çå€ 0.4771212547)
log5 = 837206/1197771 = 0.6989700034 (çå€ 0.6989700043)
log7 = 1012234/1197771 = 0.8450981031 (çå€ 0.8450980400)
log11 = 1247350/1197771 = 1.0413927203 (çå€ 1.0413926852)
log13 = 1334249/1197771 = 1.1139433164 (çå€ 1.1139433523)
log17 = 1473796/1197771 = 1.2304488922 (çå€ 1.2304489214)
log19 = 1531654/1197771 = 1.2787536182 (çå€ 1.2787536010)
log23 = 1631038/1197771 = 1.3617277426 (çå€ 1.3617278360)
log29 = 1751618/1197771 = 1.4623980711 (çå€ 1.4623979979)
忝ã®1197771ã¯10000000ãŸã§ã§æã誀差ãå°ãªããªãå€ã§ãã
ã®æ§ã«æ§æå¯èœã§ããããšã«é©ããŸãããããµãšåæ¯ãæããªããšã
忝ã¯ãã®ã«ãã£ãŠå€åãããŠããããªãã©ããªãã®ãæ°ã«ãªã£ãŠèª¿ã¹ãŠã¿ãŸããã
3æ¡ã»ã©ã®åæ°ã§ã®è¿äŒŒã¯ã忝ãæããããšã«æããªããªã
gp > abs(146/485-log(2)/log(10))
%469 = 9.3217107035117801368259509460209827810 E-7
gp > abs(73/153-log(3)/log(10))
%470 = 2.9282868735104173903973984794620415878 E-6
gp > abs(339/485-log(5)/log(10))
%471 = 9.3217107035117801368259509460209517576 E-7
gp > abs(431/510-log(7)/log(10))
%472 = 7.9857055620241233702400874249978544429 E-10
gp > abs(478/459-log(11)/log(10))
%473 = 1.6503537575300559002466218995055742675 E-6
gp > abs(743/667-log(13)/log(10))
%474 = 3.2382107964776722479812223847580763836 E-7
gp > abs(299/243-log(17)/log(10))
%475 = 3.7535188454130236161139021156448952272 E-6
gp > abs(656/513-log(19)/log(10))
%476 = 1.1643056554722575810391097558286690920 E-6
gp > abs(1103/810-log(23)/log(10))
%477 = 5.5904413551619395128281053884536869710 E-7
gp > abs(525/359-log(29)/log(10))
%478 = 2.4547234686221517882671475279717504814 E-6
ã®æ§ãªåæ°ã§ããªãã®ç²ŸåºŠãäžããããã§ãã
è¿äŒŒåæ°ã«é¢ããŠã¯ä»¥åå°ãç ç©¶ããããšããããŸãã
ïŒåæ°ã®æé ã§ç²ŸåºŠé ã«åæ°ãåæããæ¹æ³ãèããŸãã(äœåæ¡ã§ãOK)ãïŒ
æžãããŠããåæ°ã¯ããã¹ãŠé£åæ°ãæã¡åã£ãŠåŸãããåæ°ã§ããã
ãããããã¯ã3æ¡ä»¥äžã§æãè¯ããåæ°ãåŸããããšã¯éããŸããã
äŸãã°log17ã¯299/243ãã881/716ã®æ¹ãè¯ãè¿äŒŒã«ãªããŸãã
åæ§ã«log29ã525/359ãã914/625ã®æ¹ãå°ãã ãè¯ãè¿äŒŒã«ãªããŸãã
log23ã¯3æ¡ä»¥äžã§é£åæ°æã¡åãã§åŸãããåæ°ã§ã¯64/47ãæå€§ã§
粟床ãåºãªãããã«åå4æ¡ã蚱容ãããã®ãšæããŸããã
975/716ã§ãããããã®ç²ŸåºŠã¯åºãŸãã
倧åã®å€ã¯ãå°æ°ç¹ä»¥äžã®ç²ŸåºŠã(åæ¯ã®æ¡æ°Ã2)æ¡çšåºŠã«ãªããŸããã
ããŸããŸé£åæ°æã¡åãçŽåŸã®å€ã倧ããå Žåã¯ç²ŸåºŠãè¯ããªããŸããã
log7ã¯[0;1,5,2,5,6,1,4813,1,1,âŠ]ã§4813ã®åã§æã¡åã£ãŠãããã
ããã ãç¹å¥ã«ç²ŸåºŠãè¯ããªã£ãŠããŸãã
ååšçã®355/113ãåæ§ã§ããã
åã« 1 ã€ã®å¯Ÿæ°å€ãæ©æ¢°èšç®ãèš±ããŠèªç±ã«æçæ°è¿äŒŒããã ãã§ãããã
æ°åŠæåç§è©± > 环ä¹ã®äž 4 æ¡
ä»ããã®ãµã€ãã®äœã¶æãã§åãè°è«ãç¹°ãè¿ãè¡ãããŠããŸããã
29ãŸã§ã®çŽ æ°ã§ãåžžçšå¯Ÿæ°ã®é£åæ°å±éãæ±ããŠã¿ãŸããããlog_{10}(7)ã®ãšãã®4813ã®ãããªå€§ããªæ°ã¯çŸããŸããã§ããããªããlog_{10}(5)=1-log_{10}(2)ãªã®ã§çç¥ããŠããŸãã
log_{10}(2)=[0;3,3,9,2,2,4,6,2,1,1,3,1,18,...]
[0;3,3,9,2,2,4,6,2,1,1,3,1]=97879/325147
=0.301029995663499893894146339963
log_{10}(3)=[0,2,10,2,2,1,13,1,7,18,...]
[0,2,10,2,2,1,13,1,7]=34367/33001
=0.477121254550546065527863343601
log_{10}(11)=[1;24,6,3,2,1,1,3,1,1,1,9,...]
[1;24,6,3,2,1,1,3,1,1,1]=22014/21139
=1.04139268507014938941244204721
log_{10}(13)=[1;1,8,1,3,2,7,1,6,16,...]
[1;1,8,1,3,2,7,1,6]=5113/4590
=1.11394335511982570806100217865
log_{10}(17)=[1,4,2,1,17,1,13,1,1,3,3,26,...]
[1;4,2,1,17,1,13,1,1,3,3]=99797/81106
=1.23045150790323773826843883313
log_{10}(19)=[1;3,1,1,2,2,1,3,2,2,1,4,1,1,1,6,1,3,1,3,1,47,...]
[1;3,1,1,2,2,1,3,2,2,1,4,1,1,1,6,1,3,1,3,1]=6497723/5081294
=1.27875360095282815755199364571
log_{10}(23)=[1;2,1,3,4,17,2,1,2,66,...]
[1;2,1,3,4,17,2,1,2]=9016/6621
=1.36172783567436943059960731007
log_{10}(29)=[[1;2,6,6,1,2,1,2,2,2,1,1,1,1,1,5,1,2,3,37,...]
[1;2,6,6,1,2,1,2,2,2,1,1,1,1,1,5,1,2,3]=5243915/3585833
=1.46239799789895402267757589380
log[10]7ã§é£åæ°ã®8çªç®ã®å€ã4813ã§ããã
log[10]2ã¯137çªç®ã5393
log[10]3ã¯562çªç®ã2788
log[10]11ã¯2179çªç®ã3864
log[10]13ã¯133çªç®ã1378
log[10]17ã¯710çªç®ã3301
log[10]19ã¯1341çªç®ã2249
log[10]23ã¯921çªç®ã2695
log[10]29ã¯352çªç®ã1901
ã®ããã«ãã£ãšå
ãŸã§èŠãã°å€§ããªæ°ã¯ãããåºãŠããŸãã
log[10]7ã¯å¥è·¡çã«åã®æ¹ã«ãã£ããšããããšã§ããã
â ABC=ΞãšããŠã
AH=x=6tanΞ=4tan(3Ï/4-Ξ)
tanΞ=x/6ãtan(3Ï/4-Ξ)=x/4ãªã®ã§ã
tan(3Ï/4-Ξ)=(tan(3Ï/4)-tanΞ)/(1+tan(3Ï/4)tanΞ)ããã
x/4=(-1-(x/6))/(1-x/6)
ã§ãããããããã
x/4-x^2/24=-1-x/6
x^2-10x-24=0ãããªãã¡ãã
(x+2)(x-12)=0
ãã£ãŠãx=-2,12
x>0ãªã®ã§AH=x=12
åãåé¡ã«ã€ããŠãªã®ã§ãéåãããŸãã
å¹³é¢å¹Ÿäœçè§£æ³ã§ãã
ãã®äžè§åœ¢ã®å€å¿ã O ãšãããšãâ³OBC 㯠OB = OC ã§ããçŽè§äºç蟺äžè§åœ¢ãªã®ã§ã
ããšã¯ OH ã察è§ç·ãšããé·æ¹åœ¢ãæžããŠãªãããããããã°
AH = 5 + â{ (5â2)^2 - 1 } = 12
ãšæ±ãŸããŸããã
å
æ¥ãç§ã¯ãµãš 4374 ãš 4375 ãã©ã¡ãã 1 æ¡ã®çŽ å æ°ããæããªãããšã«æ°ã¥ããŸããã
ãããŠã224 ãš 225ã2400 ãš 2401 ãåæ§ã®æ§è³ªãæã€ãšç¥ã£ãŠããç§ã¯ã以äžã®ãããªèšç®ãåŸãŸããã
log ã¯å
šãŠåžžçšå¯Ÿæ°ã§ãã
224 â 225 ãã
5 log 2 + log 7 â 2 log 3 + 2 log 5
2400 â 2401 ãã
5 log 2 + log 3 + 2 log 5 â 4 log 7
4374 â 4375 ãã
log 2 + 7 log 3 â 4 log 5 + log 7
ãããŠã
log 2 + log 5 = 1
ããããé£ç«ã㊠4 å
1 次æ¹çšåŒãšæã£ãŠè§£ããšãå°æ°ç¬¬6äœåæšäºå
¥ã§
log 2 â 72/239 â 0.30126 ïŒçå€ 0.30103ïŒ
log 3 â 114/239 â 0.47699 ïŒçå€ 0.47712ïŒ
log 5 â 167/239 â 0.69874 ïŒçå€ 0.69897ïŒ
log 7 â 202/239 â 0.84519 ïŒçå€ 0.84510ïŒ
ããããŠæ¯èŒçç°¡åã«ããè¿äŒŒå€ãåŸãããããã§ãã
ãããèŠãŠçåãããã€ãã
(1)
忝 239 ãšããã®ã¯ã©ã®ãããã®åªç§ããªãã§ããããïŒ
ããªãã¡ã4 ã€ã®å¯Ÿæ°ã忝ãå
±éãªæçæ°ã§è¿äŒŒããå Žåã忝<1000 ãããã§äœçªç®ãããã«åªç§ãªè¿äŒŒå€ãåŸããã忝ãªãã§ããããïŒ
ïŒçµ¶å¯Ÿèª€å·®ã®åã§è©äŸ¡ãããçžå¯Ÿèª€å·®ã®åã§è©äŸ¡ãããã§ãå€ãããšæããŸããïŒ
(2)
4 æ¡ã§å·®ã 1 ã§ãããã®ãããã7æ¡ä»¥äžã§å·®ã 11 ã 13 ãããã¯ããããå«ã 2 æ¡ãããã®åææ°ã§ãããã®ãçšããæ¹ã粟床ããããªããããªæ°ãããŸãâŠâŠæ¬åœã§ããããïŒ
æ¬åœã ãšããŠãå
·äœçã«ã©ã®ããã粟床ãäžããããã§ãããïŒ
ïŒå·®ã 1 æ¡ã®çŽ å æ°ããæããªããã®ã¯ãABCäºæ³ã®èšŒæãä¿¡ãããªã 44100 ããå
ã«ã¯ååšããªãã¯ãïŒ
(3)
â ã§ã¯ãªããäžçå·ã§ã®è©äŸ¡ã¯åæ§ã®æ¹æ³ã§å¯èœã§ããããïŒ
(4)
䜿ãçŽ æ°ã« 11 ãå«ã㊠5 å
1 次ã«ããã䜿ãçŽ æ°ã« 13 ãŸã§å«ã㊠6 å
1 次ã«ããããªã©ã§ç²ŸåºŠã®åäžã¯å¯èœã§ããããïŒ
ç¹ã« (1) (2) (4) ã¯æäœæ¥ã§ã¯ç¡è¬ã«ãã»ã©ãããã®ã§ãã³ã³ãã¥ãŒã¿ç³»ã®æŽè»ããé¡ãããŸãã
ãšãããã(1)ã ã
> 忝 239 ãšããã®ã¯ã©ã®ãããã®åªç§ããªãã§ããããïŒ
忝ïŒ1000ã§ã¯(çžå¯Ÿèª€å·®ã®åèšã§)70çªç®ã«åªç§ã§ããã
239ã§çžå¯Ÿèª€å·®ã®åèšã¯0.001457âŠã§ãã
1äœã¯568ã§ãçžå¯Ÿèª€å·®ã®åèšã¯0.0001758âŠã§ãã
2äœä»¥äžã¯897,960,807,794,âŠãšç¶ããŸããã忝ã倧ãããã°çžå¯Ÿèª€å·®ãå°ããã®ã¯åœç¶ã§ã
ããããæå³ã§ã¯69çªç®ãŸã§ã«åæ¯ã239æªæºã®ãã®ã¯ãããŸããã®ã§ã239ã¯çµæ§åªç§ãšèšãããšæããŸãã
忝ã®å€§ãããèæ
®ããŠãçžå¯Ÿèª€å·®ã®åèšÃ忝ãã§ã©ã³ãã³ã°ãäœããšã
1äœã®568ã2äœã®897ã¯å€ãããŸãããã3äœã329ããããŠ4äœã239ãšãªããŸãã
çžå¯Ÿèª€å·®ã®åèšÃ忝ã®å
·äœå€(5äœãŸã§)ã¯
568 0.099882607730
897 0.222871229356
329 0.285662258393
239 0.348293385746
103 0.383956736568
ã®ããã«ãªã£ãŠããŠããããèŠãŠã568ã ãçªåºããŠããæãã§ãã
ã¡ãªã¿ã«åæ¯ã568ã®å Žåã®å¯Ÿæ°ã®è¿äŒŒå€ã¯
log2 â 171/568 â 0.30106
log3 â 271/568 â 0.47711
log5 â 397/568 â 0.69894
log7 â 480/568 â 0.84507
ãªã®ã§ããªãè¯ãè¿äŒŒã«ãªã£ãŠããŸããã
忝ã568ã«ãªããããªçµåããé©åœã«æ¢ããŠã¿ããšã
(2400,2401),(4374,4375),(250000,250047)
ããåŒãç«ãŠãã°äžèšã®å€ã«ãªãããã§ãã
(æ€ç®ããŠããŸããã)
ãããŒã568 åªç§ã§ããã
ããã 250047 ã¯äººåããæµç³ã«ã¡ãã£ãšåºãŠããªãâŠâŠã
ãã£ã±ãæ¡æ°ãå€ããš 2 æ°ã®å·®ãå°ããã£ãŠãæ°ã«ãªããªããªã£ãŠããã®ã§ç²ŸåºŠäžããã£ãœãã§ããã
(3)ã«ã€ããŠ
5log2 + log7 â 2log3 + 2log5
5log2 + log3 + 2log5 â 4log7
log2 + 7log3 â 4log5 + log7
ã
5log2 + log7 + a = 2log3 + 2log5
5log2 + log3 + 2log5 + b = 4log7
log2 + 7log3 + c = 4log5 + log7
ïŒa,b,cïŒ0ïŒ
ãšããŠèšç®ãããš
log2 = (72 - 27a - 5b - 7c) / 239
log3 = (114 + 17a + 12b - 31c) / 239
log5 = (167 + 27a + 5b + 7c) / 239
log7 = (202 - 16a + 59b - 13c) / 239
ãšãªããŸãããã®åŒãã
log2 ïŒ 72/239
log5 ïŒ 167/239
ã¯ãã ã¡ã«ããããŸãããlog3 ãš 114/239 ã®å€§å°é¢ä¿ã¯
17a + 12b - 31c ã®ç¬Šå·ã調ã¹ãªããšããããŸããã
ããã 17a + 12b - 31c ã®ç¬Šå·ã調ã¹ãããã«èšç®ãããš
17a + 12b - 31c = 239log3 - 114log10
ãšãªã£ãŠ 3^239 ãš 10^114 ã®å€§å°é¢ä¿ã調ã¹ãããšã«ãªããæ¬æ«è»¢åã§ãã
ãã£ãŠåæ§ã®æ¹æ³ã§äžçåŒã§äžäžããããããããã«ã¯
è¿äŒŒåŒã倿°çšæããŠããŸããŸå€§å°é¢ä¿ããããããšã«æåŸ
ããããããã
æãã€ããŸãããã远å ã®è¿äŒŒåŒãçšæããããšãããšæ¡æ°ãå¢ããŠ
æèšç®ã«äžåãã«ãªã£ãŠããŸãã®ã§ããšãããã
ããã®æ¹æ³ã§ã®äžçå·ã§ã®è©äŸ¡ã¯é£ããã
ãšèšã£ãŠããããšæããŸãã
> 17a + 12b - 31c ã®ç¬Šå·ã調ã¹ãªããšããããŸããã
ãªãã»ã©ãå·®åã宿°åããŠããŸãã°ããã£ãã®ã§ããã
é£ç«æ¹çšåŒãè§£ãæéã¯ãã£ããå¢ããŠããŸããŸãããã©ãã
æã§ãããªãéè¡åçšæããŠè§£ãã®ãäžçªæ©ãããªïŒ
{1/n - 1/(2*n^2)} log e < log{1+(1/n)} < {1/n} log e
ã䜿ãã°ãlog e ã¯æ¬ãåºããŠæŸçœ®ã§ããã®ã§ãa, b, c ã®ç·åçµåã®æ£è² è©äŸ¡ã¯ãªããšããªãã±ãŒã¹ãå€ããã«æããŸãã
(224, 225), (2400, 2401), (4374, 4375) ã®ã±ãŒã¹ã¯å®éããã§ãªããšããªãã¿ããã§ãã
é·æã§ãã
(2)ã«ã€ããŠ
çŽ æ°2,3,5,7ã10æ¡ä»¥äžã§(2400,2401)ãã誀差çãå°ãããã®ã¯
以äžã®6åãããããŸããã§ããã巊端ã¯èª€å·®ç(倧ããæ¹ã®å€Ã·å°ããæ¹ã®å€ïŒ1)ã§ãã
0.000040616 78121827 78125000
0.000066758 645657712 645700815
0.000107377 3954653486 3955078125
0.000188000 250000 250047
0.000228624 4374 4375
0.000295397 184473632 184528125
0.000416667 2400 2401
78121827 = 3^13 * 7^2, 78125000 = 2^3 * 5^10, å·® = 3173 = 19 * 167
645657712 = 2^4 * 7^9, 645700815 = 3^17 * 5, å·® = 43103 (çŽ æ°)
3954653486 = 2 * 7^11, 3955078125 = 3^4 * 5^11, å·® = 424639 (çŽ æ°)
ããã䜿ã£ãŠ
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
4log2 + 9log7 = 17log3 + log5 (645657712, 645700815)
log2 + 11log7 = 4log3 + 11log5 (3954653486, 3955078125)
log2 + log5 = 1
ãè§£ããšäžæ¬¡åŸå±ã§è§£ããŸããã§ããã
ãããããŠ
78121827 = 3^13 * 7^2, 78125000 = 2^3 * 5^10, å·® = 3173 = 19 * 167
645657712 = 2^4 * 7^9, 645700815 = 3^17 * 5, å·® = 43103 (çŽ æ°)
250000 = 2^4 * 5^6, 250047 = 3^6 * 7^3, å·® = 47 (çŽ æ°)
ããã䜿ã£ãŠ
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
4log2 + 9log7 = 17log3 + log5 (645657712, 645700815)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + log5 = 1
ãè§£ããš
log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
ããã¯èŠèŠãããããŸããã
ããããã®å€ã¯(2400,2401),(4374,4375),(250000,250047)
ãããåŸãããã®ã§ã¯ãªãããšæã£ãŠäžã§ãæ€ç®ããŠããŸããããšæžããã®ã
ãããããŠæ€ç®ããŠã¿ããšããªããš
5log2 + log3 + 2log5 = 4log7 (2400, 2401)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + log5 = 1
ã¯äžæ¬¡åŸå±ã§è§£ããŸããã§ãããçµæ§è§£ããªãå ŽåãåºãŠããã®ã§ããã
ããã§ã¯ããããçµåããå€ããŠè©Šãããšã«ããŸãã
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
log2 + 11log7 = 4log3 + 11log5 (3954653486, 3955078125)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + log5 = 1
â log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
4log2 + 9log7 = 17log3 + log5 (645657712, 645700815)
log2 + 11log7 = 4log3 + 11log5 (3954653486, 3955078125)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + log5 = 1
â log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
4log2 + 9log7 = 17log3 + log5 (645657712, 645700815)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
log2 + log5 = 1
â log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
log2 + 11log7 = 4log3 + 11log5 (3954653486, 3955078125)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
log2 + log5 = 1
â log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
log2 + log5 = 1
â äžæ¬¡åŸå±
4log2 + 9log7 = 17log3 + log5 (645657712, 645700815)
4log2 + 6log5 = 6log3 + 3log7 (250000, 250047)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
log2 + log5 = 1
â log2 = 171/568, log3 = 271/568, log5 = 397/568, log7 = 480/568
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
5log2 + log3 + 2log5 = 4log7 (2400, 2401)
log2 + log5 = 1
â äžæ¬¡åŸå±
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ãããã10æ¡ä»¥äžãã11æ¡ä»¥äžã12æ¡ä»¥äžãã»ã»ã»ãšå¢ãããŠããªããªãèŠã€ãããŸããã
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0.000026141 205885750000000 205891132094649
0.000033563 281474976710656 281484423828125
ã®äºã€ãèŠã€ãããŸããã®ã§ããããš(78121827, 78125000)ã§è©ŠããŸãã
205885750000000 = 2^7 * 5^9 * 7^7, 205891132094649 = 3^30,
å·® = 5382094649 = 3673 * 1465313
281474976710656 = 2^48, 281484423828125 = 5^11 * 7^8,
å·® = 9447117469 (çŽ æ°)
7log2 + 9log5 + 7log7 = 30log3 (205885750000000, 205891132094649)
48log2 = 11log5 + 8log7 (281474976710656, 281484423828125)
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
log2 + log5 = 1
â
log2 = 3125/10381 = 0.30103073 (çå€ 0.30103000)
log3 = 4953/10381 = 0.47712166 (çå€ 0.47712125)
log5 = 7256/10381 = 0.69896927 (çå€ 0.69897000)
log7 = 8773/10381 = 0.84510163 (çå€ 0.84509804)
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0.000007053 2251783932057135 2251799813685248
2251783932057135 = 3^13 * 5 * 7^10, 2251799813685248 = 2^51,
å·® = 15881628113 = 13 * 71 * 17206531
ã䜿ã£ãŠ
13log3 + log5 + 10log7 = 51log2 (2251783932057135, 2251799813685248)
7log2 + 9log5 + 7log7 = 30log3 (205885750000000, 205891132094649)
48log2 = 11log5 + 8log7 (281474976710656, 281484423828125)
log2 + log5 = 1
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äœ¿ãæ¡æ°ãäžãã£ãŠãã2, 3, 5, 7 ã§äœããåææ°ã®å²åãæžãããšã§æã¡æ¶ãããŠããŸãã誀差çããªããªãå°ãããªããªããã§ããã
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ã11ã远å ããå Žåã
(1äžãŸã§)
0.000102041 9800 9801
0.000228624 4374 4375
0.000330688 3024 3025
0.000416667 2400 2401
0.001244444 5625 5632
9800 = 2^3 * 5^2 * 7^2, 9801 = 3^4 * 11^2
4374 = 2 * 3^7, 4375 = 5^4 * 7
3024 = 2^4 * 3^3 * 7, 3025 = 5^2 * 11^2
2400 = 2^5 * 3 * 5^2, 2401 = 7^4
3log2 + 2log5 + 2log7 = 4log3 + 2log11 (9800, 9801)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
4log2 + 3log3 + log7 = 2log5 + 2log11 (3024, 3025)
5log2 + log3 + 2log5 = 4log7 (2400, 2401)
log2 + log5 = 1
â äžæ¬¡åŸå±
5625 = 3^2 * 5^4, 5632 = 2^9 * 11, å·® = 7
3log2 + 2log5 + 2log7 = 4log3 + 2log11 (9800, 9801)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
5log2 + log3 + 2log5 = 4log7 (2400, 2401)
2log3 + 4log5 = 9log2 + log11 (5625, 5632)
log2 + log5 = 1
â
log2 = 270/897 = 0.301003 (çå€ 0.301030)
log3 = 428/897 = 0.477146 (çå€ 0.477121)
log5 = 627/897 = 0.698997 (çå€ 0.698970)
log7 = 758/897 = 0.845039 (çå€ 0.845098)
log11 = 934/897 = 1.041249 (çå€ 1.041393)
897ã¯äžã®æ¹ã§568ã«æ¬¡ãã§ç²ŸåºŠã®è¯ã忝ã§ãã
(1åãŸã§)
0.000016089 3294172 3294225
0.000022158 67108864 67110351
0.000040616 78121827 78125000
0.000050668 14348180 14348907
3294172 = 2^2 * 7^7, 3294225 = 3^2 * 5^2 * 11^4, å·® = 53 (çŽ æ°)
67108864 = 2^26, 67110351 = 3 * 7^5 * 11^3, å·® = 1487 (çŽ æ°)
78121827 = 3^13 * 7^2, 78125000 = 2^3 * 5^10, å·® = 3173 = 19 * 167
14348180 = 2^2 * 5 * 7^2 * 11^4, 14348907 = 3^15, å·® = 727 (çŽ æ°)
2log2 + 7log7 = 2log3 + 2log5 + 4log11 (3294172, 3294225)
26log2 = log3 + 5log7 + 3log11 (67108864, 67110351)
13log3 + 2log7 = 3log2 + 10log5 (78121827, 78125000)
2log2 + log5 + 2log7 + 4log11 = 15log3 (14348180, 14348907)
log2 + log5 = 1
â
log2 = 6421/21330 = 0.3010314 (çå€ 0.3010300)
log3 = 10177/21330 = 0.4771214 (çå€ 0.4771213)
log5 = 14909/21330 = 0.6989686 (çå€ 0.6989700)
log7 = 18026/21330 = 0.8451008 (çå€ 0.8450980)
log11 = 22213/21330 = 1,0413971 (çå€ 1.0413927)
çµæ§ç²ŸåºŠãäžãããŸããã
ã13ã远å ããå Žåã
(1äžãŸã§)
0.000102041 9800 9801
0.000150263 6655 6656
0.000228624 4374 4375
0.000236742 4224 4225
0.000244200 4095 4096
0.000330688 3024 3025
0.000416667 2400 2401
9800 = 2^3 * 5^2 * 7^2, 9801 = 3^4 * 11^2
6655 = 5 * 11^3, 6656 = 2^9 * 13
4374 = 2 * 3^7, 4375 = 5^4 * 7
4224 = 2^7 * 3 * 11, 4225 = 5^2 * 13^2
4095 = 3^2 * 5 * 7 * 13, 4096 = 2^12
3024 = 2^4 * 3^3 * 7, 3025 = 5^2 * 11^2
2400 = 2^5 * 3 * 5^2, 2401 = 7^4
3log2 + 2log5 + 2log7 = 4log3 + 2log11 (9800, 9801)
log5 + 3log11 = 9log2 + log13 (6655, 6656)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
7log2 + log3 + log11 = 2log5 + 2log13 (4224, 4225)
2log3 + log5 + log7 + log13 = 12log2 (4095, 4096)
log2 + log5 = 1
â äžæ¬¡åŸå±
3log2 + 2log5 + 2log7 = 4log3 + 2log11 (9800, 9801)
log5 + 3log11 = 9log2 + log13 (6655, 6656)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
7log2 + log3 + log11 = 2log5 + 2log13 (4224, 4225)
2log3 + log5 + log7 + log13 = 12log2 (4095, 4096)
4log2 + 3log3 + log7 = 2log5 + 2log11 (3024, 3025)
log2 + log5 = 1
â äžæ¬¡åŸå± (åŒãäžã€å€ããŠããªãäžæ¬¡åŸå±ãªã®ã§ä»ã®åŒãå¿
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3log2 + 2log5 + 2log7 = 4log3 + 2log11 (9800, 9801)
log5 + 3log11 = 9log2 + log13 (6655, 6656)
log2 + 7log3 = 4log5 + log7 (4374, 4375)
7log2 + log3 + log11 = 2log5 + 2log13 (4224, 4225)
5log2 + log3 + 2log5 = 4log7 (2400, 2401)
log2 + log5 = 1
â
log2 = 270/897 = 0.301003 (çå€ 0.301030)
log3 = 428/897 = 0.477146 (çå€ 0.477121)
log5 = 627/897 = 0.698997 (çå€ 0.698970)
log7 = 758/897 = 0.845039 (çå€ 0.845098)
log11 = 934/897 = 1.041249 (çå€ 1.041393)
log13 = 999/897 = 1.113712 (çå€ 1.113943)
11ã ã远å ãããšããšåã粟床ã§ãã
(1åãŸã§)
0.000007456 5767125 5767168
0.000008117 123200 123201
0.000013783 72772425 72773428
0.000015573 1990625 1990656
0.000016089 3294172 3294225
0.000018861 19140264 19140625
0.000022158 67108864 67110351
5767125 = 3 * 5^3 * 7 * 13^3, 5767168 = 2^19 * 11, å·® = 43 (çŽ æ°)
123200 = 2^6 * 5^2 * 7 * 11, 123201 = 3^6 * 13^2
72772425 = 3^7 * 5^2 * 11^3, 72773428 = 2^2 * 7^2 * 13^5, å·® = 1003 = 17 * 59
1990625 = 5^5 * 7^2 * 13, 1990656 = 2^13 * 3^5, å·® = 31 (çŽ æ°)
3294172 = 2^2 * 7^7, 3294225 = 3^2 * 5^2 * 11^4, å·® = 53 (çŽ æ°)
19140264 = 2^3 * 3^2 * 11^2 * 13^3, 19140625 = 5^8 * 7^2, å·® = 361 = 19^2
67108864 = 2^26, 67110351 = 3 * 7^5 * 11^3, å·® = 1487 (çŽ æ°)
log3 + 3log5 + log7 + 3log13 = 19log2 + log11 (5767125, 5767168)
6log2 + 2log5 + log7 + log11 = 6log3 + 2log13 (123200, 123201)
7log3 + 2log5 + 3log11 = 2log2 + 2log7 + 5log13 (72772425, 72773428)
5log5 + 2log7 + log13 = 13log2 + 5log3 (1990625, 1990656)
2log2 + 7log7 = 2log3 + 2log5 + 4log11 (3294172, 3294225)
log2 + log5 = 1
â äžæ¬¡åŸå±
log3 + 3log5 + log7 + 3log13 = 19log2 + log11 (5767125, 5767168)
6log2 + 2log5 + log7 + log11 = 6log3 + 2log13 (123200, 123201)
7log3 + 2log5 + 3log11 = 2log2 + 2log7 + 5log13 (72772425, 72773428)
2log2 + 7log7 = 2log3 + 2log5 + 4log11 (3294172, 3294225)
3log2 + 2log3 + 2log11 + 3log13 = 8log5 + 2log7 (19140264, 19140625)
log2 + log5 = 1
â
log2 = 6079/20194 = 0.3010300089 (çå€ 0.3010299957)
log3 = 9635/20194 = 0.4771219174 (çå€ 0.4771212547)
log5 = 14115/20194 = 0.6989699911 (çå€ 0.6989700043)
log7 = 17066/20194 = 0.8451025057 (çå€ 0.8450980400)
log11 = 21030/20194 = 1.0413984352 (çå€ 1.0413926852)
log13 = 22495/20194 = 1.1139447361 (çå€ 1.1139433523)
粟床ã¯11ã ã远å ã®ãšããšåçšåºŠã§ãã
ïŒlog2ãšlog5ã¯ç²ŸåºŠãããã§ãããä»ã¯ãããŸã§è¯ããããŸããïŒ
ã29ãŸã§è¿œå ããå ŽåïŒçŽ æ°10åïŒã
(1åãŸã§)
0.000000010 96059600 96059601
0.000000055 18085704 18085705
0.000000075 26578123 26578125
0.000000084 11859210 11859211
0.000000095 10556000 10556001
0.000000121 8268799 8268800
0.000000155 12901779 12901781
0.000000169 5909760 5909761
0.000000194 5142500 5142501
0.000000244 4096575 4096576
0.000000244 4090624 4090625
0.000000250 4004000 4004001
0.000000315 22194425 22194432
0.000000365 13697019 13697024
0.000000365 90312467 90312500
0.000000371 2697695 2697696
0.000000485 8254125 8254129
0.000000489 88012332 88012375
0.000000494 2023424 2023425
0.000000520 90312453 90312500
0.000000540 1852200 1852201
0.000000560 67874587 67874625
0.000000569 75557027 75557070
0.000000587 46000759 46000786
96059600 = 2^4 * 5^2 * 7^2 * 13^2 * 29, 96059601 = 3^8 * 11^4
18085704 = 2^3 * 3 * 7^3 * 13^3, 18085705 = 5 * 11 * 17 * 23 * 29^2
26578123 = 11 * 13^2 * 17 * 29^2, 26578125 = 3^5 * 5^6 * 7
11859210 = 2 * 3^4 * 5 * 11^4, 11859211 = 7 * 13 * 19^4
10556000 = 2^5 * 5^3 * 7 * 13 * 29, 10556001 = 3^4 * 19^4
8268799 = 7^2 * 11 * 23^2 * 29, 8268800 = 2^10 * 5^2 * 17 * 19
12901779 = 3^2 * 11 * 19^4, 12901781 = 23^2 * 29^3
5909760 = 2^8 * 3^5 * 5 * 19, 5909761 = 11^2 * 13^2 * 17^2
5142500 = 2^2 * 5^4 * 11^2 * 17, 5142501 = 3^3 * 7^2 * 13^2 * 23
4096575 = 3^4 * 5^2 * 7 * 17^2, 4096576 = 2^6 * 11^2 * 23^2
4090624 = 2^8 * 19 * 29^2, 4090625 = 5^5 * 7 * 11 * 17
4004000 = 2^5 * 5^3 * 7 * 11 * 13, 4004001 = 3^2 * 23^2 * 29^2
22194425 = 5^2 * 11^3 * 23 * 29, 22194432 = 2^8 * 3^3 * 13^2 * 19
13697019 = 3^4 * 7^3 * 17 * 29, 13697024 = 2^16 * 11 * 19
90312467 = 7 * 23^2 * 29^3, 90312500 = 2^2 * 5^7 * 17^2
2697695 = 5 * 7^3 * 11^2 * 13, 2697696 = 2^5 * 3^2 * 17 * 19 * 29
8254125 = 3^2 * 5^3 * 11 * 23 * 29, 8254129 = 13^4 * 17^2
88012332 = 2^2 * 3^4 * 17 * 19 * 29^2, 88012375 = 5^3 * 11^3 * 23^2
2023424 = 2^13 * 13 * 19, 2023425 = 3^2 * 5^2 * 17 * 23^2
90312453 = 3^2 * 7 * 11 * 19^4, 90312500 = 2^2 * 5^7 * 17^2
1852200 = 2^3 * 3^3 * 5^2 * 7^3, 1852201 = 13 * 17^3 * 29
67874587 = 11^2 * 23 * 29^3, 67874625 = 3^3 * 5^3 * 7 * 13^2 * 17
75557027 = 7 * 13^3 * 17^3, 75557070 = 2 * 3^3 * 5 * 23^4
46000759 = 7^6 * 17 * 23, 46000786 = 2 * 13^3 * 19^2 * 29
4log2 + 2log5 + 2log7 + 2log13 + log29 = 8log3 + 4log11 (96059600, 96059601)
3log2 + log3 + 3log7 + 3log13 = log5 + log11 + log17 + log23 + 2log29 (18085704, 18085705)
log11 + 2log13 + log17 + 2log29 = 5log3 + 6log5 + log7 (26578123, 26578125)
log2 + 4log3 + log5 + 4log11 = log7 + log13 + 4log19 (11859210, 11859211)
2log7 + log11 + 2log23 + log29 = 10log2 + 2log5 + log17 + log19 (8268799, 8268800)
2log3 + log11 + 4log19 = 2log23 + 3log29 (12901779, 12901781)
8log2 + 5log3 + log5 + log19 = 2log11 + 2log13 + 2log17 (5909760, 5909761)
4log3 + 2log5 + log7 + 2log17 = 6log2 + 2log11 + 2log23 (4096575, 4096576)
6log7 + log17 + log23 = log2 + 3log13 + 2log19 + log29 (46000759, 46000786)
log2 + log5 = 1
â
log2 = 360565/1197771 = 0.3010299966 (çå€ 0.3010299957)
log3 = 571482/1197771 = 0.4771212527 (çå€ 0.4771212547)
log5 = 837206/1197771 = 0.6989700034 (çå€ 0.6989700043)
log7 = 1012234/1197771 = 0.8450981031 (çå€ 0.8450980400)
log11 = 1247350/1197771 = 1.0413927203 (çå€ 1.0413926852)
log13 = 1334249/1197771 = 1.1139433164 (çå€ 1.1139433523)
log17 = 1473796/1197771 = 1.2304488922 (çå€ 1.2304489214)
log19 = 1531654/1197771 = 1.2787536182 (çå€ 1.2787536010)
log23 = 1631038/1197771 = 1.3617277426 (çå€ 1.3617278360)
log29 = 1751618/1197771 = 1.4623980711 (çå€ 1.4623979979)
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There are eight gold coins, one of which is a forgery containing radioactive material. The task is to identify this forgery using a series of measurements conducted by technicians with Geiger counters. The problem is structured as follows:
1. Coins: There are 8 gold coins, numbered 1 through 8. Exactly one coin is a forgery.
2. Forgery Characteristics: The forged coin contains radioactive material, detectable by a Geiger counter.
3. Technicians: There are 10 technicians available to perform measurements.
4. Measurement Process:
Each technician selects a subset of the 8 coins for measurement.
The technician uses a Geiger counter to test the selected coins simultaneously.
The Geiger counter reacts if and only if the forgery is among the selected coins.
Only the technician operating the device knows the result of the measurement.
5. Measurement Constraints:
Each technician performs exactly one measurement.
A total of 10 measurements are conducted.
6. Reporting:
After each measurement, the technician reports either "positive" (radioactivity detected) or "negative" (no radioactivity detected).
7. Reliability Issue: Up to two technicians may provide unreliable reports, either due to intentional deception or unintentional error.
8. Objective: Identify the forged coin with certainty, despite the possibility of up to two unreliable reports.
Challenge
The challenge is to design a measurement strategy and analysis algorithm that can definitively identify the forged coin, given these constraints and potential inaccuracies in the technicians' reports.
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onst codes = [
"100 111 010 ", // XORçµæ 001
"111 010 110 ", // XORçµæ 011
"010 110 001 ", // XORçµæ 101
"110 001 011 ", // XORçµæ 100
"001 011 101 ", // XORçµæ 111
"011 101 100 ", // XORçµæ 010
"101 100 111 ", // XORçµæ 110
];
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z = 000
ãšããŸãããŸãã
a = 100
b = 111
c = 010
d = 110
e = 001
f = 011
g = 101
ãšããŸãã
ãã®ãšã以äžã®æ§è³ªããããŸãã(ããã§ã¯âãæä»çè«çåãšããŸãã)
e = aâbâc
f = bâcâd
g = câdâe
a = dâeâf
b = eâfâg
c = fâgâa
d = gâaâb
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x = 011, y = 101 ã®ãšã
xâ¥y = 011101 ãšãªããŸãã
e, f, g, a, b, c, d, z ããšã³ã³ãŒãã㊠10 ãããã«ãããã®ããããããE, F, G, A, B, C, D, Z ãšåä»ããŸãã
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E = aâ¥bâ¥câ¥p(c)
F = bâ¥câ¥dâ¥p(d)
G = câ¥dâ¥eâ¥p(e)
A = dâ¥eâ¥fâ¥p(f)
B = eâ¥fâ¥gâ¥p(g)
C = fâ¥gâ¥aâ¥p(a)
D = gâ¥aâ¥bâ¥p(b)
Z = zâ¥zâ¥zâ¥p(z)
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E=100â¥111â¥010â¥1 âe=001
F=111â¥010â¥110â¥0 âf=011
G=010â¥110â¥001â¥1 âg=101
A=110â¥001â¥011â¥0 âa=100
B=001â¥011â¥101â¥0 âb=111
C=011â¥101â¥100â¥1 âc=010
D=101â¥100â¥111â¥1 âd=110
Z=000â¥000â¥000â¥0 âz=000
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0,0,0,0,0,0,0,0,0,0
1,0,0,1,1,0,0,1,1,1
0,1,0,1,0,1,0,1,1,0
1,1,0,0,1,1,0,0,0,1
0,0,1,1,0,0,1,1,0,1
1,0,1,0,1,0,1,0,1,0
0,1,1,0,0,1,1,0,1,1
1,1,1,1,1,1,1,1,0,0
ãšããã³ãŒããèããããŸãã10äººã®æè¡è
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0|0,0,0,0,0,0,0,0,0,0 â8çªãåœé å
1|1,0,0,1,1,0,0,1,1,1 â1çªãåœé å
2|0,1,0,1,0,1,0,1,1,0 â2çªãåœé å
3|1,1,0,0,1,1,0,0,0,1 â3çªãåœé å
4|0,0,1,1,0,0,1,1,0,1 â4çªãåœé å
5|1,0,1,0,1,0,1,0,1,0 â5çªãåœé å
6|0,1,1,0,0,1,1,0,1,1 â6çªãåœé å
7|1,1,1,1,1,1,1,1,0,0 â7çªãåœé å
10äººã®æè¡è
ã®å ±åã®ãã¡2人ã®å ±åã誀ãã§ããå Žåã¯ã
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"111 010 110 001 011 101 100",
"010 110 001 011 101 100 111",
"110 001 011 101 100 111 010",
"001 011 101 100 111 010 110",
"011 101 100 111 010 110 001",
"101 100 111 010 110 001 011",
"000 000 000 000 000 000 000",
Minimum Hamming Distance: 12
ãªããäžèšã§ã¯ 21 äžã«ã誀ããªãæè¡è
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A,B,C,D,E,F,G,H,I,J,K,L,M
-------------------------
0,0,0,0,0,0,0,0,0,0,0,0,0
1,0,0,1,1,0,1,0,0,1,1,0,1
0,1,0,1,0,1,0,1,0,1,0,1,1
1,1,0,0,1,1,1,1,0,0,1,1,0
0,0,1,0,1,1,0,0,1,0,1,1,1
1,0,1,1,0,1,1,0,1,1,0,1,0
0,1,1,1,1,0,0,1,1,1,1,0,0
1,1,1,0,0,0,1,1,1,0,0,0,1
13äººã®æè¡è
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A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q
---------------------------------
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
1,0,0,1,0,0,1,0,0,1,1,1,0,1,1,1,0
0,1,0,0,1,0,0,1,0,1,1,0,1,1,1,0,1
1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1
0,0,1,0,0,1,0,0,1,1,0,1,1,1,0,1,1
1,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,1
0,1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0
1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0
17äººã®æè¡è
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A,B,C,D,E,F
-----------
0,0,0,0,0,0
1,0,0,1,1,0
0,1,0,1,0,1
1,1,0,0,1,1
0,0,1,0,1,1
1,0,1,1,0,1
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[
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]
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