1.ïŒæã®ãšãŒã¹ãéžã³ããŒãã«ã«è¡šåãã«äžŠã¹ãŠåºãã
2.ãã®ããããã«ä»»æã®è£åãã§ïŒæã®ã«ãŒããèŒããã
3.客ã«èµ€ããšãŒã¹ã®ïŒã€ã®ãã±ãããéããŠæž¡ããèªç±ã«æãã£ããã·ã£ããã«ãããã
ãïŒïŒæã®ã«ãŒãã®äžã«ïŒæã®è¡šåãã®ãšãŒã¹ãã©ã®äœçœ®ã«ããŠãæ§ããªããïŒ
ãããªããé»ã®ãšãŒã¹ã®ã«ãŒã矀ãæãã客ãšåæ§èªç±ã«ã·ã£ããã«ããã
4.客ãæž¡ããã±ãããå³æã«ãã¡ãå·Šæã«ã¯èªåãã·ã£ããã«ãããã±ãããã²ã£ããè¿ããŠ
ãä¿æãããïŒïŒæã®è¡šåãã«ãŒããšïŒæã®è£åãã®ãšãŒã¹ã«ãŒãã®ç¶æ
ãïŒ
ãïŒå®¢ã®ãã±ããã¯ïŒæãè£åãã§ãïŒæã®ãšãŒã¹ã¯è¡šåããïŒ
5.äž¡æã«æã€ãã±ããã®äžããããŒãã«ã«äº€äºã«äžæãã€ã®ã«ãŒãéããªãã眮ããŠããã
ãïŒè¡šåãã ã£ãããè£åãã ã£ããããã«ãŒããéãªã£ãŠãããïŒ
6.åºãçµãã£ãã«ãŒãã®æãæŽããŠã客ã«ä»»æã®å Žæã§ã«ãããããã
ã(ã«ããã¯ã·ã£ããã«ãšã¯ç°ãªããã«ãŒãå
šäœã§ã®åç°çé åºãå€ããªããïŒ
7.ãããåãåãããŒãã«ã«ïŒåã«äžæãã€åºããŠéããŠãããïŒåå±±ïŒæãã€ãïŒ
8.ïŒã€ã®å±±ã«åºæ¥ãã«ãŒãã®æã§ãå³ç«¯ã®å±±ããã¹ãŠã²ã£ããè¿ãããã®å·Šé£ã®å±±ã«éããã
ããããŠããã®éããå±±ã®æå
šäœããã¹ãŠã²ã£ããè¿ãããŸããã®å·Šé£ã®å±±ã«éããã
ããã®ããšãããäžåºŠç¹°ãè¿ããïŒã€ãã£ãå±±ãäžã€ã®ã«ãŒãã®æã«ããã
9.ãã®éãªã£ããã±ãããããŒãã«ã«ãªãã³ã¹ãã¬ããããæãäœãèµ·ãããã¯ãèªèº«ã§
ã確ãããŠäžããã
27000001ã®å æ°å解ã§
27000001=27000000+1
=300^3+1^3
ããã§a^3+b^3=(a+b)^3-3*a*b*(a+b)=(a+b)*((a+b)^2-3*a*b)
ãã3*a*bã®éšåãå¹³æ¹æ°ãšãªãå Žåã§
ãã®äŸã§ã
=301*(301^2-3*300*1)
=301*(301^2-30^2)
=301*(271)*(331)
=7*43*271*331
ãã®æ§ãª3ä¹ã®åïŒN=a^3+b^3 ããã3*a*bãå¹³æ¹æ°ïŒ
ã§ãäžèšã®ã«ãŒãã§çŽ å æ°å解ã§ããã¿ã€ãã®æ°Nã
10^7å°ã«éã£ãŠèª¿æ»ããŠã¿ãŸããã(å
šéšã§89å)
(2*10^7å°ã§ã¯27000001ãåœç¶çŸããã)
N [a , b]=æçµã®å æ°å解圢
10021508[213,71]=2^2*7*71^3
10063872[192,144]=2^12*3^3*7*13
10077704[216,2]=2^3*7*13*109*127
10078208[216,8]=2^11*7*19*37
10083528[216,18]=2^3*3^6*7*13*19
10110464[216,32]=2^9*7^2*13*31
10202696[216,50]=2^3*7*19*43*223
10450944[216,72]=2^11*3^6*7
10706059[196,147]=7^7*13
10892476[219,73]=2^2*7*73^3
11018888[216,98]=2^3*31*157*283
11313512[224,42]=2^3*7^4*19*31
11346272[222,74]=2^5*7*37^3
11375000[200,150]=2^3*5^6*7*13
11390652[225,3]=2^2*3^3*7*13*19*61
11392353[225,12]=3^3*7^2*79*109
11410308[225,27]=2^2*3^6*7*13*43
11501217[225,48]=3^3*7*13*31*151
11812500[225,75]=2^2*3^3*5^6*7
11859211[228,19]=7*13*19^4
11904697[192,169]=7^2*19^2*673
12071241[204,153]=3^3*7*13*17^3
12110644[189,175]=2^2*7^4*13*97
12174848[216,128]=2^9*7*43*79
12291328[228,76]=2^8*7*19^3
12650337[225,108]=3^6*7*37*67
12782924[231,77]=2^2*7^4*11^3
12795328[208,156]=2^6*7*13^4
13287456[234,78]=2^5*3^3*7*13^3
13547807[212,159]=7*13*53^3
13805092[237,79]=2^2*7*79^3
13824125[240,5]=5^3*7^2*37*61
13832000[240,20]=2^6*5^3*7*13*19
13915125[240,45]=3^3*5^3*7*19*31
14172704[242,6]=2^5*7*13*31*157
14186312[242,24]=2^3*7*19*67*199
14329224[216,162]=2^3*3^9*7*13
14329952[242,54]=2^5*7^2*13*19*37
14336000[240,80]=2^14*5^3*7
14348908[243,1]=2^2*7*31*61*271
14348971[243,4]=7*13*19*43*193
14349636[243,9]=2^2*3^6*7*19*37
14353003[243,16]=7*37*151*367
14364532[243,25]=2^2*7*13*19*31*67
14395563[243,36]=3^6*7^2*13*31
14466556[243,49]=2^2*13*37*73*103
14567148[225,147]=2^2*3^3*19*31*229
14607424[196,192]=2^6*13*97*181
14611051[243,64]=7*13*307*523
14709500[245,15]=2^2*5^3*13*31*73
14880348[243,81]=2^2*3^12*7
14922125[245,60]=5^3*19*61*103
15057224[242,96]=2^3*7*13^2*37*43
15140125[220,165]=5^3*7*11^3*13
15348907[243,100]=7^3*73*613
15438304[246,82]=2^5*7*41^3
15652000[250,30]=2^5*5^3*7*13*43
15777125[240,125]=5^3*7*13*19*73
15981056[224,168]=2^9*7^4*13
16010036[249,83]=2^2*7*83^3
16012269[252,21]=3^3*7^4*13*19
16120468[243,121]=2^2*7*13*67*661
16595712[252,84]=2^8*3^3*7^4
16777243[256,3]=7*37*211*307
16778944[256,12]=2^6*7*13*43*67
16796899[256,27]=7*61*139*283
16852563[228,171]=3^3*7*13*19^3
16887808[256,48]=2^12*7*19*31
17166500[245,135]=2^2*5^3*13*19*139
17195500[255,85]=2^2*5^3*7*17^3
17199091[256,75]=7*13*331*571
17334891[243,144]=3^6*7*43*79
17353000[250,120]=2^3*5^3*7*37*67
17547488[242,150]=2^5*7^2*19^2*31
17755192[232,174]=2^3*7*13*29^3
17809568[258,86]=2^5*7*43^3
18036928[256,108]=2^6*7*13*19*163
18077696[216,200]=2^11*7*13*97
18410392[264,22]=2^3*7*11^3*13*19
18438084[261,87]=2^2*3^3*7*29^3
18468513[225,192]=3^3*7*19*37*139
18689489[236,177]=7*13*59^3
19081216[264,88]=2^11*7*11^3
19175716[243,169]=2^2*7*61*103*109
19656000[240,180]=2^6*3^3*5^3*7*13
19684000[270,10]=2^5*5^3*7*19*37
19739132[267,89]=2^2*7*89^3
19747000[270,40]=2^3*5^3*7^2*13*31
19953739[256,147]=13*31*67*739
ãªã倧åŠå
¥è©Šã«æèšç®ã§æ¬¡ã®æ°ãçŽ å æ°å解ããããã®ã
åºé¡ãããŠããŸããã
N=12345654321
N0=110001011
N1=11111111
N2=11112121
N3=133113133
N4=14141441
N5=15151515115
N6=11611661
N7=17171111
N8=1811811818
N9=191111911
ã¯æèšç®ã§å æ°å解ã§ãããã®ãªã®ãïŒ
ãšããããç¬æ®ºã§ãããã®ããã
N1 = 11111111 = 1111*10001 = 11*101*10001
ããŠãGAI ãããããã§çµããã ãã®é¢çœã¿ã®ãªãåé¡ãåºããšã¯æããªãã®ã§ã10001 ã¯åææ°ãããããããã倧ããªçŽ æ°ã®ç©ã ãšä¿¡ããããšã«ããŸãã
10001 ã 2 ã€ã®èªç¶æ°ã®ç©ã§æžããšèãããšããã®çžä¹å¹³å㯠â10001 㧠100 ãããããã«å€§ããæ°ã§ãã
ãŸãã10001 㯠4 ã§å²ããš 1 äœãæ°ãªã®ã§ãããã 2 ã€ã®æ°ã®ç©ã§ãããªãã°ãã㯠4 ã§å²ããš 1 äœãæ°å士ã®ç©ããããã㯠4 ã§å²ããš 3 äœãæ°å士ã®ç©ã
ã€ãŸãããã® 2 ã€ã®æ°ã®å㯠4 ã§å²ããš 2 äœããŸãã
ããã 2 ã€ã®æ
å ±ã«ãã©ã¡ãããããã倧ããªçŽ å æ°ãšããæ
å ±ãè¿œå ãããšã2 æ°ã®çžå å¹³å㯠100 ããå°ã倧ããå¥æ°ã§ãããšããããŸãã
ãšããããšã§ãããã 101+2k ãšæžãããšã«ããŸãã
ãããš 2 ã€ã®æ°ã解ã«æã€äºæ¬¡æ¹çšåŒã¯
x^2 - 2(101+2k)x + 10001 = 0
ãšãªãããã®å€å¥åŒã¯
D/4 = (101+2k)^2 - 10001 = 4k^2 + 404k + 200 = 4(k^2+101k+50)
ããšã¯ãã®æ¬åŒ§å
ãå¹³æ¹æ°ã«ãªããã㪠k ã®å€ãå°ããé ã«è©Šããªããæ¢ãã°ããã
k=1 ã®ãšã 152 ã¯å¹³æ¹æ°ã§ã¯ãªã
k=2 ã®ãšã 256 ã¯å¹³æ¹æ°
ãšããã«ã¿ã€ãããŸãã
2 æ°ã®çžå å¹³åã 105 ãšããããšã¯å㯠210 ã§ãç©ã 10001 ãªã®ã§ããããå·®ã¯
â(210^2-4*10001) = â4096 = 64
ã€ãŸã 2 æ°ã¯ 105 + 32 = 137 ãš 105 - 32 = 73
以äžãããN1 = 11111111 = 11*73*101*137
åãããæ¹ã§ãã 1 ã€ã
N2 = 11112121 = 11111111+1010
ãšèãããšãN1 ã®çµæãšåãããŠããã 101 ã®åæ°ã§ããããšã¯æããã§ã
N2 = 11112121 = 101*110021
éå¹³æ³ã䜿ã£ãŠé 匵ãã° â110021â331.7 ã§ããšããããšã¯ 2 æ°ã®çžå å¹³å㯠331 ããå°ã倧ããå¥æ°ãªã®ã§ 331+2k ãšãããŠãåæ§ã«é²ããŠã
D/4 = (331+2k)^2 - 110021 = 4(k^2+331k-115)
ãã®æ¬åŒ§å
ãå¹³æ¹æ°ã«ãªã k ãæ¢ããŸãã
k=1 ã®ãšã 217 ã¯å¹³æ¹æ°ã§ã¯ãªã
k=2 ã®ãšã 551 ã¯å¹³æ¹æ°ã§ã¯ãªã
k=3 ã®ãšã 887 ã¯å¹³æ¹æ°ã§ã¯ãªã
k=4 ã®ãšã 1225 ã¯å¹³æ¹æ°
2 æ°ã®çžå å¹³åã 339 ã§ãåã 678ãç©ã 110021 ãªã®ã§ãå·®ã¯
â(678^2-4*110021) = â19600 = 140
ã€ãŸã 2 æ°ã¯ 339 + 70 = 409 ãš 339 - 70 = 269
以äžãã N2 = 11112121 = 101*269*409
äžå¿ 19 以äžã®çŽ æ°ã§å²ã£ãŠã¿ãŠããããå
šéšçŽ æ°ãšç¢ºèªããŠçµäºã
N7 ããããããšæããŸãããã170011 ã®åŠçããã®æ¹æ³ã§ã¯ç¡çããã§ããã
2 ã€ã®æ°ãããããå以äžå·®ãããããã§ããã®æ¹æ³ã§ã¯ã¡ãã£ãšå³ããã
ããã©ããããããªã
f(k)=(412+2k)^2-170011ãšãããš
f(k)=4k^2+1648k-267
kãå¶æ°ã®ãšãf(k)â¡5(mod8)ãšãªãã
mod8ã§ã®å¹³æ¹å°äœã¯0,1,4ã ããªã®ã§å¹³æ¹æ°ã«ãªããªãã
k=2m-1ãšãããšf(k)=g(m)=16m^2+3280m-1911
mâ¡0,1,2,3,4,5,6,7,8ã«å¯ŸããŠg(m)â¡6,8,6,0,8,3,3,8,0(mod9)ã ã
mod9ã§ã®å¹³æ¹å°äœã¯0,1,4,7ã ããªã®ã§
å¹³æ¹æ°ã«ãªãå¯èœæ§ãããã®ã¯mâ¡3,8(mod9)ã®ãšãã®ã¿ã
mâ¡3(mod9)ã®ãšãm=9t-6ãšãããš
g(m)=h(t)=1296t^2+27792t-21015
h(1)=8073, h(2)=39753ã¯äžã®äœã3ãªã®ã§å¹³æ¹æ°ã§ã¯ãªãã
h(3)=74025ãå¹³æ¹æ°ãªãã°27^2=729,28^2=784ããh(3)=275^2ã§ãªããã°
ãªããªããã275^2=75625ãªã®ã§h(3)ã¯å¹³æ¹æ°ã§ã¯ãªãã
h(4)=110889ãå¹³æ¹æ°ãªãã°33^2=1089,34^2=1156ãã
h(4)=333^2ãŸãã¯337^2ã§ãªããã°ãªããªããã333^2=110889ãªã®ã§
h(4)ã¯å¹³æ¹æ°ã
ïŒããŸããŸèŠã€ãã£ãã®ã§mâ¡8(mod9)ã¯èããå¿
èŠããªããªã£ãïŒ
t=4âm=30âk=59â412+2k=530ãªã®ã§
170011=530^2-333^2ãšãããã以äžç¥ã
mod 10ãã§ã¯
a^5â¡a
ãæºããaã¯{1,2,3,4,5,6,7,8,9}ãšãã«æ°åã§ããã£ãã
ãããmod 100ãã«ãããš
{1,7,24,25,32,43,49,51,57,68,75,76,93,99}
ã§ãããã§ããç©è¶³ããªãã
ããã§
a^kâ¡a (mod 100)ãå€ãã®aãéããããkã¯åŠäœã«ïŒ
ã§èª¿æ»ããŠã¿ããšäœãš
gp > for(a=1,99,if(lift(Mod(a^21,100))==a,print1(a",")))
1,3,4,7,8,9,11,12,13,16,17,19,21,23,24,25,
27,28,29,31,32,33,36,37,39,41,43,44,47,48,49,
51,52,53,56,57,59,61,63,64,67,68,69,71,72,73,75,
76,77,79,81,83,84,87,88,89,91,92,93,96,97,99,
ãã®åãA075821ã«èŒãã(å
容çã«ã¯ä»ã®èŠç¹ã§éãŸã£ãæ°åïŒ
62/99(çŽ62.6ïŒ
)ãã®ãã®ãæ¡çšå¯èœãšãªãããã³ããã§ãã£ãã
äžæè°ãªããšã«kã¯ä»ã®ã
k=41,61,81,101,
ã§ãåãaã䞊ãã ã
ãŸãmod 1000
ã§ã¯k=101,201,301,
ããã§éãŸãaã®å²åãã504/999=56/111(çŽ50.5ïŒ
ïŒã§ãã³ããã§ããã
ããã§æ¡çšãããaã®å€ãããããšã¯å
šãããé¢ããå
容ã§ã®A122987
ãšäžèŽããããšã«é©ããã
ãã®ïŒã€ã®é¢ä¿ã¯ã©ããªã£ãŠãããã ããïŒ
> ãã®åãA075821ã«èŒãã(å
容çã«ã¯ä»ã®èŠç¹ã§éãŸã£ãæ°åïŒ
ã³ã¡ã³ããèªãéãããŸããã 21 ä¹ã®äž 2 æ¡ãšããŠäœãããåã®ããã§ããã
> äžæè°ãªããšã«kã¯ä»ã®ã
k=41,61,81,101,
ã§ãåãaã䞊ãã ã
> ãŸãmod 1000
ã§ã¯k=101,201,301,
ãã£ãšãå
æ¥ã®åé¡ã®æµããåããã°ãã®çµæã¯ããèªç¶ãªãã®ã«æããŸãããã©ããäžæè°ãšæã£ãŠããããã®ã§ããããïŒ
a41=a21*a20â¡a*a^20=a21â¡aãïŒmod 100)
ãããäžæè°ã§ããªãã§ããªãã§ããã
A122987ã§ã®èª¬æãããããããªãã®ã§ãããããã
a^101â¡a (mod 1000)
ã§éããaãšåãã«ãªãã®ã¯ã©ãããŠãªã®ããªïŒ
ãšããæå³ã§åããããŠãããŸããã
ãªãã»ã©ç¢ºãã«ç«æ¹æ°ã®äž 3 æ¡ãšã®äžèŽã¯å°ãèããªããšãããŸãããã
以äžã§ã©ãã§ãããã
以äžãåååŒã¯æãããªãéã mod1000 ãšããŸãã
101 ä¹ã®äž 3 æ¡ãå
ã®æ°ã«æ»ãæ°ã¯ãå¿
ãããç«æ¹æ°ã®äž 3 æ¡ã«åºçŸããŸãã
ãªããªãã°ãa^101â¡a ã§ããã°ãbâ¡a^67 ãšãããšã
b^3â¡a^201â¡a^101â¡a ã ããã§ãã
ç«æ¹æ°ã®äž3æ¡ã«åºçŸããæ°ã¯ããã®101ä¹ã®äž3æ¡ãå
ã®æ°ã«äžèŽããŸãã
ãªããªãã以äžã® 2 ã€ãããç«æ¹æ°ãšäž 3 æ¡ãäžèŽãã a ã«ã€ã㊠a^101-a 㯠1000 ã®åæ°ã ããã§ãã
(1) ç«æ¹æ°ã¯ 5 ãšäºãã«çŽ ã§ããããŸãã¯ããèªäœ 125 ã®åæ°ã§ãã
ãã£ãŠ a ãåãã 5 ãšäºãã«çŽ ã§ããããŸãã¯ããèªäœ 125 ã®åæ°ã§ãã
ã€ãŸããa^Ï(125)-1 ããªãã¡ a^100-1 ãŸã㯠a ã®ããããã 125 ã®åæ°ã§ãã
ãããã£ãŠãa(a^100-1) 㯠125 ã®åæ°ã§ãã
(2) ç«æ¹æ°ã¯å¥æ°ã§ããããŸãã¯ããèªäœ 8 ã®åæ°ã§ãã
ãã£ãŠ a ãåããå¥æ°ã§ããããŸãã¯ããèªäœ 8 ã®åæ°ã§ãã
ã€ãŸããa^2-1 ãŸã㯠a ã®ããããã 8 ã®åæ°ã§ãã
ãããã£ãŠãa(a^100-1)=a(a^2-1)(a^98+a^96+âŠâŠ+a^2+1) 㯠8 ã®åæ°ã§ãã
a^21â¡a (mod 100)
ãæãaã®çžåœæ°ãèŠã€ãããããããæºããaã®éåã¯
Mod(a^3,100)ã§ã®äœããåãåŸãæ°ã«å¯Ÿå¿ã
a^101â¡a(mod 1000)
ãæãaã®çžåœæ°ãèŠã€ãããããããæºããaã®éåã¯
Mod(a^3,1000)ã§ã®äœããåãåŸãæ°ã«å¯Ÿå¿ããŠããã
ïŒããã°ã©ã ã§ã®ç¢ºèªïŒ
gp > {S=[];}for(a=1,99,r=lift(Mod(a^3,10^2));S=concat(S,[r]));S=vecsort(Set(S))
%41 = [0, 1, 3, 4, 7, 8, 9, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99]
gp > {T=[];}for(a=0,99,if(lift(Mod(a^21,10^2))==a,T=concat(T,[a])));T
%42 = [0, 1, 3, 4, 7, 8, 9, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99]
çµæã¯é·ããªãã®ã§çç¥ããŠããŸãããçµæã¯åãã«ãªããŸããã
gp > {S=[];}for(a=1,999,r=lift(Mod(a^3,10^3));S=concat(S,[r]));S=vecsort(Set(S))
gp > {T=[];}for(a=0,999,if(lift(Mod(a^101,10^3))==a,T=concat(T,[a])));T
ããã§
mod 10000
ã調ã¹ãŠã¿ãã
a^501â¡a (mod 10000)
ãæãaã®çžåœæ°ãèŠã€ããã4509åãããïŒãã³ããã®å€ãïŒ
({T=[];}for(a=0,9999,if(lift(Mod(a^501,10^4))==a,T=concat(T,[a])));T
ã§æ±ãŸãéåTã®èŠçŽ æ°#Tã#T=4509)
ãšããã
a^3ã10000ã§å²ã£ããšãã®äœããåãåŸãç·æ°ã¯5050åãšãªã
({S=[];}for(a=1,9999,r=lift(Mod(a^3,10^4));S=concat(S,[r]));S=vecsort(Set(S))
ã§æ±ãŸãéåSã§ã®#S=5050)
äžïŒã€ã®åºããã¯èµ·ãããŸããã§ããã
ããããã§ããã
mod100 ã®å Žå㯠a^n ã® n ãšããŠã2 以äžã〠20 ãšäºãã«çŽ ãã§ããããšãèŠæ±ãããã®ã§ n=3 ãæå°
mod1000 ã®å Žå㯠a^n ã® n ãšããŠã3 以äžã〠100 ãšäºãã«çŽ ãã§ããããšãèŠæ±ãããã®ã§ n=3 ãæå°
mod10000 ã®å Žå㯠a^n ã® n ãšããŠã4 以äžã〠500 ãšäºãã«çŽ ãã§ããããšãèŠæ±ãããã®ã§ n=7 ãæå°
ãšãªãã®ã§ã¯ãªãã§ãããããïŒ
{S=[];}for(n=1,9999,r=lift(Mod(n^7,10^4));S=concat(S,[r]));S=vecsort(Set(S))
{T=[];}for(n=0,9999,if(lift(Mod(n^501,10^4))==n,T=concat(T,[n])));T
ã«å¯ŸããŠ#S=#T=4509
ããã
gp > S==T
% = 1 (SãT ã®ïŒã€ã®éåå
容ãå
šãåäžã瀺ããïŒ
ã®çµæãšãªããDD++ããã®æšæž¬ã¯èŠäºã«å®èšŒã§ããŸããã
ã¡ãªã¿ã«
mod 100000ã§ã¯
{S=[];}for(n=1,99999,r=lift(Mod(n^7,10^5));S=concat(S,[r]));S=vecsort(Set(S))
{T=[];}for(n=0,99999,if(lift(Mod(n^5001,10^5))==n,T=concat(T,[n])));T
ã®å¯Ÿå¿ã§ïŒã€ã®éåã¯åäžãèŠãŸããã(#S=#T=42517)
äŸãã°ãå®çŸ©åãæçæ°å
šäœã§ããé¢æ°f(x)=x^2ãã¯åŸ®åäžå¯èœãšãããŠãããšæããŸããã
ããã¯xãæçæ°äžã§åãããŠæ¥µéããšãããšã§åŸ®åå¯èœãšå®çŸ©ããŠãåé¡ãªãããã«æããŸãã
ïŒäžè¬ã«ã¯ãæçæ°ã§ãªããŠãçš å¯ã§ããã°ããïŒ
ãªã埮åäžå¯èœãšå®çŸ©ãããã®ã§ãããïŒ
# ã©ããã®ãµã€ãã«ãç¡çæ°ã®ãšãã«å¥ã®å€ãå®çŸ©ãããšåŸ®åäžå¯èœã«ãªãããããš
# æžãããŠããŸããããããã¯ãtan(Ï/2)=0ãšå®çŸ©ãããštanã埮åäžå¯èœã«ãªãããšåããªã®ã§
# ããŸãçç±ã«ãªããªããšæããŸãã
å°ãèããŠã¿ãŸããããããããã®åé¡ã¯ãæçæ°ã®ã¿ãšãå€æ°ã§æ¥µéãèããŠãããã©ããããªã®ã§ã¯ãªãã§ããããã
ãããŠãããã NO ã§ãããããã極éãå®çŸ©ã§ããªãâé£ç¶æ§ã®å€å®ãã§ããªãâé£ç¶é¢æ°ã§ãªãã®ã ããåœç¶åŸ®åäžå¯èœããšããçå±ã«ãªã£ãŠããã®ã§ã¯ãªãããšæããŸãã
ãšããã®ããä»®ã«æçæ°ã®ã¿ãšãå€æ°ã®æ¥µéãèããŠããããšã«ãããšãäžéå€ã®å®çããæ倧å€æå°å€ã®å®çãããã®ä»è«žã
ã®å®çããŽããœãªäžæç«ã«ãªã£ãŠããŸããã§ãããã
ãã¡ããšæã£ãäžã§æçæ°å€æ°ã®æ¥µéãå°å
¥ããã°äœãããããçè«äœç³»ãã§ãããã§ããã倱ããã®ã®å€§ããã®ããã«ã¡ãªããã¯ã»ãšãã©ãªãããã
åçããããšãããããŸãããªãã»ã©ã確ãã«ããããåé¡ãããããã§ããã
ãããšããããããšãããå¥ã®ååã§ç°ãªãçè«ãšããŠäœç³»ãäœããªããã°ãªããªããã§ããããããããã®ãèŠãããšããªãããšãããäœç³»ãäœã£ãŠã䜿ãããããªãããšããããšãªã®ã§ããããã
mod 10ãã§
a^5â¡a
ãæºããaã¯{1,2,3,4,5,6,7,8,9}
ã ã£ãã®ã§ã5ä¹ã¯å
ã«æ»ãåãç¹å¥ãšæããã®ã§
ã§ã¯mod 100,mod 1000,,mod 10^nã§ã¯ã©ããªæ°åã察å¿ã§ããã
調ã¹ãŠã¿ãããšã«ããã
a^5â¡a(mod 100)
ãæºããæŽæ°aã¯
{1,7,24,25,32,43,49,51,57,68,75,76,93,99}
a^5â¡a(mod 1000)
ãæºããæŽæ°aã¯
{1,57,125,193,249,251,307,375,376,432,443,499,501,557,568,624,625,693,749,751,807,875,943,999}
ããã§ãããã系統å¥ã«
1->51->251
 ->751
->01->501
2->32->432
3->43->443
->943
->93->193
->693
4->24->624
5->25->125
->625
->75->375
->875
6->76->376
7->57->557
->07->307
->807
8->68->568
9->49->249
->749
->99->499
->999
ãšãã颚ã«åã«æºè¶³ããŠããæŽæ°ã®é ã«ãæ°ããªæ°åãä»ãå ããããšã§
ç¹ããŠããããã®ãæ¢ããŠã¿ãããšã«ããã
次ã®åè£ã¯
{1,443,624,625,807,1249,1251,1693,1875,2057,2499,2501,2943, 3125,3307,3568,3749,3751, 4193,4375,4557,4999,5001,5443,5625,5807, 6249,6251,6432,6693,6875,7057,7499,7501,7943,8125,8307,8749,8751, 9193,9375,9376,9557,9999}
ãšãªãã®ã§,ããã«ã€ãªããŠããã
ããããŠæ¬¡ã
ãšç¹ãã£ãŠãããåãã次ã®ãã®ãèŠã€ãã£ãã
ããšã¯ãããOEISã§æ€çŽ¢ãããããããã®ã®æ²èŒåãåèã«ã€ããŠããŸãã
A063006(A224474)ãã
M1=[1, 5, 7, 8, 1, 2, 4, 7, 5, 3, 6, 1, 0, 8, 4, 7, 8, 4, 5, 1,];
1,51,751,8751,18751,218751,4218751,74218751,574218751,3574218751,
ã€ãŸã
[51^5â¡51(mod 10^2),751^5â¡751(mod 10^3),8751^5â¡8751(mod 10^4),ãæç«ããã]
A120817ãã
M2=[2, 3, 4, 6, 8, 1, 9, 7, 8, 9, 9, 4, 3, 6, 2, 3, 0, 1, 4, 0,];
2,32,432,6432,86432,186432,9186432,79186432,879186432,9879186432,
A290373ãã
M31=[3, 4, 9, 2, 2, 9, 7, 0, 9, 1, 8, 5, 6, 7, 4, 0, 4, 6, 3, 0,];
3,43,943,2943,22943,922943,7922943,7922943,907922943,1907922943,
A290375ãã
M32=[3, 9, 1, 4, 0, 7, 3, 3, 3, 8, 1, 4, 6, 9, 9, 2, 5, 1, 8, 8,];
3,93,193,4193,4193,704193,3704193,33704193,333704193,8333704193,
A091664(A216092)ãã
M4=[4, 2, 6, 0, 9, 8, 2, 1, 2, 8, 1, 9, 9, 5, 2, 6, 5, 2, 2, 9,];
4,24,624,624,90624,890624,2890624,12890624,212890624,8212890624,
A091663(A216093)ãã
M51=[5, 7, 3, 9, 0, 1, 7, 8, 7, 1, 8, 0, 0, 4, 7, 3, 4, 7, 7, 0,];
5,75,375,9375,9375,109375,7109375,87109375,787109375,1787109375,
A018247(A007185)ãã
M52=[5, 2, 6, 0, 9, 8, 2, 1, 2, 8, 1, 9, 9, 5, 2, 6, 5, 2, 2, 9,];
5,25,625,625,90625,890625,2890625,12890625,212890625,8212890625,
ïŒãã ãããã¯5ä¹ã«éãããäœä¹ã§ãæç«ããŠãããïŒ
A018248(A016090)ãã
M6=[6, 7, 3, 9, 0, 1, 7, 8, 7, 1, 8, 0, 0, 4, 7, 3, 4, 7, 7, 0,];
6,76,376,9376,9376,109376,7109376,87109376,787109376,1787109376,
ïŒãã ãããã¯5ä¹ã«éãããäœä¹ã§ãæç«ããŠãããïŒ
A290372ãã
M71=[7, 0, 8, 5, 9, 2, 6, 6, 6, 1, 8, 5, 3, 0, 0, 7, 4, 8, 1, 1,];
7,7,807,5807,95807,295807,6295807,66295807,666295807,1666295807,
A290374ãã
M72=[7, 5, 0, 7, 7, 0, 2, 9, 0, 8, 1, 4, 3, 2, 5, 9, 5, 3, 6, 9,];
7,57,57,7057,77057,77057,2077057,92077057,92077057,8092077057,
A120818ãã
M8=[8, 6, 5, 3, 1, 8, 0, 2, 1, 0, 0, 5, 6, 3, 7, 6, 9, 8, 5, 9,];
8,68,568,3568,13568,813568,813568,20813568,120813568,120813568,
A091661(A224473)ãã
M9=[9, 4, 2, 1, 8, 7, 5, 2, 4, 6, 3, 8, 9, 1, 5, 2, 1, 5, 4, 8,];
9,49,249,1249,81249,781249,5781249,25781249,425781249,6425781249,
確ãã«5ä¹ã¯mod 10 ã«éããä»ã®mod 10^n ã§ã®äžçã§ãå
ã®æ°ã«åŒãæ»ãããšã
åºæ¥ã圹å²ãæ
ãç¶ããããšãåºæ¥ããã§ãã(å1ïœ9ã«ç¶ã系統ãååšããã)
N=11^100+22^100+33^100+44^100+55^100+66^100+77^100+88^100+99^100
ã10ã§å²ã£ãäœãã¯ïŒ
以äžåååŒã¯ãã¹ãŠmod10
2^4=16â¡6
3^4=81â¡1
4^2=16â¡6
5^nâ¡5
6^nâ¡6
7^4=2401â¡1
8^4=4096â¡6
9^2=81â¡1
ãã
Nâ¡1^100+2^100+3^100+4^100+5^100+6^100+7^100+8^100+9^100
=1+(2^4)^25+(3^4)^25+(4^2)^50+5^100+6^100+(7^4)^25+(8^4)^25+(9^2)^50
â¡1+6+1+6+5+6+1+6+1=33â¡3
ãªã®ã§ã3ã
äžåŒâ¡1+0+1+0+1+0+1+0+1â¡1 (mod2)
ãŸããp=5 ã«ã€ããŠãã§ã«ããŒã®å°å®çãçšãããšã
äžåŒâ¡1+1+1+1+0+1+1+1+1â¡3 (mod5)
ãã£ãŠ
äžåŒâ¡3 (mod10)
ãäºäººãšãæ§è³ªãçç¥ãããŠããããšããã³ãã³äŒãã£ãŠããŸãã
ã²ãããªããšãã5ä¹ã«ãããŠã¯,a=1,2,3,,9ã§
a^5â¡a (mod 10)
ãæç«ããŠããããšã«æ°ä»ããŠãããã掻çšã§ããåé¡ãšããŠäœæããŠãããŸããã
ç¹ã«DD++ããã®æãç°¡æœãªè¿éã«ææ¿ããŸããã
ãªããããmod 100
ãšãªã£ãå Žåã¯ã©ã®æ§ã«å¯ŸåŠã§ããã®ã§ããïŒ
> ãªããããmod 100ãšãªã£ãå Žåã¯ã©ã®æ§ã«å¯ŸåŠã§ããã®ã§ããïŒ
以äžåååŒã¯ãã¹ãŠmod100
n^2â¡nã®è§£ã¯0,1,25,76
(ã€ãŸããã®4ã€ã¯äœä¹ããŠãmod100ã§äžå€)
2^20=1048576â¡76
3^20=3486784401â¡1
4^10=1048576â¡76
5^2=25â¡25
6^5=7776â¡76
7^4=2401â¡1
8^20=1152921504606846976â¡76
9^10=3486784401â¡1
11^10=25937424601â¡1
ãªã®ã§
N=11^100+22^100+33^100+44^100+55^100+66^100+77^100+88^100+99^100
=11^100(1^100+2^100+3^100+4^100+5^100+6^100+7^100+8^100+9^100)
={(11^10)^10}{1+(2^20)^5+(3^20)^5+(4^10)^10+(5^2)^50+(6^5)^20+(7^4)^25+(8^20)^5+(9^10)^10}
â¡1ã»(1+76+1+76+25+76+1+76+1)
=333â¡33
ãšãªãã100ã§å²ã£ãäœãã¯33ã
ãªã€ã©ãŒã®ããŒã·ã§ã³ãé¢æ°ã䜿ããŸãã
ãªã€ã©ãŒã®å®çãããa ã 5 ã®åæ°ã§ãªããšã
a^Ï(25)=a^20â¡1 (mod25)
ãªã®ã§ã
äžåŒâ¡1+1+1+1+0+1+1+1+1â¡8 (mod25)
ãŸãã
äžåŒâ¡1+0+1+0+1+0+1+0+1â¡1 (mod4)
ãã£ãŠ
äžåŒâ¡33 (mod100)
ã€ãã§ã« 1000 ã§å²ãå Žåãã
ãªã€ã©ãŒã®å®çãããa ã 5 ã®åæ°ã§ãªããšã
a^Ï(125)=a^100â¡1 (mod125)
ãªã®ã§ã
äžåŒâ¡1+1+1+1+0+1+1+1+1â¡8 (mod125)
ãŸãã
äžåŒâ¡1+0+1+0+1+0+1+0+1â¡5 (mod8)
ãã£ãŠ
äžåŒâ¡133 (mod1000)
10000 ã§å²ããšãªããšæãå€ããªããšãããŸãããã
(303-n)/(320+n)=23/47
ã®è§£ã¯ 983 ã§ã¯ãªã 98.3 ã§ãããæŽæ°ã§ã¯ãªãã®ã§ã¯ã
303/320=3030/3200ããšããèŠæ¹ã¯ãéªéã§ãããïŒ
åé¡æã®è¡šçŸããååããåŒãããåæ¯ã«è¶³ãããªããã§ãã®ã§ã
åé¡ïŒ303/320 ã®ååã¯äœã
解çïŒ3030
ãæ£è§£ãšããã¹ããã©ãããšãã話ã«ãªããŸããã
ç§ã¯äžæ£è§£ãšãã¹ãã ãšæããŸããã©ãã§ãããïŒ
ãŸãããã®è«ãèªããå Žåã303/320 ã 6060/6400 ãšã¿ãªãããšã§ 1966 ãªã©å¥ã®è§£ãèªããããŠããŸããŸããã
極端ã«ã¯ 303/320 ã (
303/98.3)/(320/98.3) ãšèŠåãã° 1 ã解ã«ãªãããã®åé¡ã®çãã¯ãä»»æã®æŽæ°ãã«ãªãããšæããŸãã
(5^2+8^2-7^2)/(2*5*8)=1/2ãšããèšç®ãã
3蟺ã5,7,8ã®äžè§åœ¢ã®5ãš8ã®èŸºã§æãŸããè§ã®è§åºŠã¯60°ãšãããã
(æ¢ç¥ãšããã°èšç®äžèŠ)
å³ã®åœ¢ã¯3蟺ã5,7,8ã®äžè§åœ¢ã®5ã®èŸºãš8ã®èŸºã®å€åŽã«
ããããæ£äžè§åœ¢ããã£ã€ãã圢ãªã®ã§ãAB=5+8=13ã
å³ã§ã®
AD=a,BC=b,CD=c,AB=xãšãããŠ,ãã®4ã€ãæŽæ°ãšãªããçµåã調ã¹ãã
(a,b,c)=(1,4,7)->x=9
=(2,5,7)->x=10
=(3,6,7)->x=11
=(4,7,7)->x=12
=(5,8,7)->x=13
=(6,9,7)->x=14
=(7,10,7)->x=15
ãããã
äžè¬ã«(a,b,c)=(n,n+3,7)->x=n+8 (n=1,2,3,)
ãŸãã¯
(a,b,c)=(1,6,7)->x=9
=(2,7,7)->x=10
=(3,8,7)->x=11
=(4,9,7)->x=12
=(5,10,7)->x=13
=(6,11,7)->x=14
=(7,12,7)->x=15
ãããã
äžè¬ã«(a,b,c)=(n,n+5,7)->x=n+8 (n=1,2,3,)
c=7ãšèšå®ããŠããããšããã€ã³ãã«ãªãããã§ãã
c=13 ã c=19 ã§ãå¯èœãªã®ã§ã¯ã
ãããã 6n+1 åçŽ å æ°ãå°ãªããšã 1 ã€æã£ãŠããããšãæ¡ä»¶ãããªãã§ããããã
確ãã«6*n+1åã®æ°ã¯
7^2=3^2+8^2-3*8
=5^2+8^2-5*8
13^2=7^2+15^2-7*15 =>(a,b,c)=(n,n+8,13)->x=n+15 (n=1,2,3,)
=8^2+15^2-8*15 =>(a,b,c)=(n,n+7,13)->x=n+15 (n,1,2,3,)ãæ§æã§ããã
19^2= 5^2+21^2- 5*21 =>(a,b,c)=(n,n+16,19)->x=n+21 (n=1,2,3,)
=16^2+21^2-16*21 =>(a,b,c)=(n,n+5,19)->x=n+21 (n=1,2,3,)
25^2=25^2+25^2-25*25(ããã¯äŸå€ïŒ
31^2=11^2+35^2-11*35=>(a,b,c)=(n,n+24,31)->x=n+35 (n=1,2,3,)
=24^2+35^2-24*35=>(a,b,c)=(n,n+11,31)->x=n+35 (n=1,2,3,)
以äžåæ§
37^2= 7^2+40^2-7*40
=33^2+40^2-24*40
43^2=13^2+48^2-13*48
=35^2+48^2-35*48
49^2=16^2+55^2-16*55
=39^2+55^2-39*55

ãš60°ã®è§åºŠãæããäžè§åœ¢ã®äžèŸºãäžããŠããïŒçµã
äžããŠãããŸããã
ãããããŒãã§æ£æã«çŠè¢ãæºåããŠããããæºåããŠããçŠè¢ã®æ°ãè¶
ããŠ
éåºåã«ãããç®åœãŠã®å®¢ãå€æ°äžŠãã§ããŸã£ãã
ããã§åŸãã«äžŠãã§ãã人ã«ããã£ã³ã¹ãå·¡ã£ãŠããããã«ã次ã®ãããªæ¡ã
èããã
䞊ãã§ããå
é ãã1,2,3,ãšé£ç¶ããçªå·æãé
ã£ãŠããã
å
é ã«ãã人ã«ã¯çŠè¢ãè²·ãæš©å©ãäžãããã®ãšããã
(çªå·1ã®äººã¯è²·ããããã®äººã¯åããé¢ããã)
次ã¯2çªã®äººãå
é ã«æ¥ãã®ã§ã2çªã®äººãè²·ããã
ããã§çªå·ã2ãªã®ã§å
é ãã2çªç®ãã€ã®äœçœ®ã«ãã人
(2,3,4,5,6,7,8,9,10,11,12,)
<=4,6,8,10,12,ã®çªå·æãæã£ãŠãã人>
ã¯åããé¢ããŠãããã
ããã§åã¯
3,5,7,9,11,13,15,17,19,21,23,
ãšäžŠã¶ããšã«ãªãã®ã§ã
å
é ã¯çªå·ã3(ãã®äººã¯è²·ãæš©å©ãæã€ïŒãªã®ã§
å
é ãã3çªç®ãã€ã®äœçœ®ã«ãã人ã¯åæ§ã«åããé¢ããŠãããã
<=9,15,21,ã®çªå·æã®äºº>
ãããšåã¯
5,7,11,13,17,19,23,
ãšãªã5ã®äººã¯è²·ãæš©å©ãæã¡ãå
é ãã5çªãã€ã®äœçœ®ã«ãã人<=19,35,>
ã¯åããé¢ããã
以äžåæ§ã«ããŠãåã«äžŠã¶äººãããªããªããŸã§ç¶ããããšã«ããã
ããŠæå䞊ãã§ãã人æ°ã100,1000,10000(人)ã§ããå Žå
ããããã¯çŠè¢äœå(s)æºåããŠããã°ããããŸã
æåŸã«è²·ããæš©å©ãæã€ã®ã¯çªå·æãäœçª(w)ã®äººã«ãªãã§ããããïŒ
çè«çã«èšç®ããæ¹æ³ã¯ããããŸããã§ããã®ã§ã
ããã°ã©ã ãäœã£ãŠèª¿ã¹ãŸããããã®çµæã¯
人æ°100,1000,10000,100000,1000000,10000000,100000000人ã«å¯ŸããŠ
(s,w)=(24, 97),(142, 997),(1015, 9997),(7986, 99997),
(66164, 999991),(565513, 9999985),(4944199, 99999967)
ãšãªããŸããã
管ç人ããããã®ã³ã¡ã³ãã§
åçŽã«ããåŒãã§ãïŒïŒäººã®åœéžè
ã決ããŠããã£ãæ¹ãã䞊ã¶äººã®ææ
ãšããŠã¯çŽ
åŸã§ãããšæãã®ã ãïŒïŒïŒã
ãšãããŸãããããã£ããæ©ã䞊ãã§ãã1,2,3ãããã®äººãç±€ã§åœãããªãã£ãããããããäžæºãæºãŸãããã§ãã
ããã ãšæ¯èŒçæ©ã䞊ãã æ¹ã®äººãéžã°ããããåŸåãæãŠãæ°ãããã®ã§ããã®ã¹ããŒãªãŒã§è¡šçŸããŠããŸããã