> äžç¹ã質åããããŠããã ãããæããŸãã
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No.1116ãããã2023幎5æ20æ¥ 20:12
> åæ§ã«ã2ãš3ã§ç¢ºçççŽ æ°ãšãªãã°6ã§ãïŒå€åïŒç¢ºçççŽ æ°ãšå€å®ããããšæããŸãã
ç§ã¯ããã¯ãªããšãªãåœã§ã¯ãªãããšæããŸãã
åäŸã¯ãŸã èŠã€ããããŠããŸãããã
No.1117DD++2023幎5æ21æ¥ 10:59
> 2ãš3ã§ç¢ºçççŽ æ°ãšãªãã°6ã§ãïŒå€åïŒç¢ºçççŽ æ°
äŸãã°nãçŽ æ°ã§n-1=16mïŒmã¯å¥æ°ïŒã ãšãããšãmod nã§
(a^m,a^(2m),a^(4m),a^(8m),a^(16m))â¡
(1,1,1,1,1),
(-1,1,1,1,1),
(A,-1,1,1,1),
(B,C,-1,1,1),
(D,E,F,-1,1)
(A,B,C,D,E,Fã¯-1,0,1以å€ã®æ°)
ã®ããããã«ãªãããã§ããã
a=2ã§
(A,-1,1,1,1)
a=3ã§
(D,E,F,-1,1)
ã®ããã«ãªã£ããšãããšãa=6ã§ã¯
(A*D,-E,F,-1,1)ãïŒA*Dã-EãFã-1,0,1以å€ïŒ
ã®ããã«ãªãããã¯ã確çççŽ æ°ãšå€å®ãããŸãã
åé¡ã¯
åº2㧠(A,B,C,-1,1)
åº3㧠(D,E,F,-1,1)
ã®ããã«åãã¿ã€ãã³ã°ã§-1ã«ãªã£ãå Žåã§ã
ãã®ãããªå Žåã¯C*Fâ¡-1ã«ãªããšã¯éããŸããã®ã§
åº6ã§ç¢ºçççŽ æ°ãšå€å®ããããã©ããããããŸããã
ïŒããã®åäŸãããã®ãã©ãããããããŸããããããããªæ°ã¯ããŸããïŒ
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ãšããããšãèšãããã£ãã®ã§ãã
No.1118ãããã2023幎5æ21æ¥ 12:35
> å°ãªããšã-1ã«ãªãã¿ã€ãã³ã°ãç°ãªãçŽ å æ°ã®ç©ãåºã«ããå€å®ã¯ç¢ºçã®åäžã«ãªããŸããã®ã§ãåãçŽ å æ°ã¯é¿ããæ¹ãããã
ãããå®ã¯å°ãèããªããã°ãããªããšããã§ã¯ãªãããšæããŸãã
äŸãã° 13 ããã©ãŒã©ãã³ãã¹ãã«ããããšããŸãã
ãã®ãšããa = 3, 7, 11 ãšãã 3 ã€ã®çŽ æ°ã§å€å®ãè¡ãããšã¯æå¹ã§ããããïŒ
æ®éã®èªç¶æ°ã®äžçã§ã¯ã7 ã¯çŽ æ°ã§ãã
ãããã13 ããã©ãŒã©ãã³ãã¹ãã«ãããå Žåãè¡ãèšç®ã¯å
šãŠ mod 13 ã®äžçã§ãã
ãã®äžçã§ã¯ã3*11=7 ã§ãããã7 ã¯ãããããããä»°ããšããã®é¿ããæ¹ãããæ°ã«è©²åœããŸãã
ããèããŠã¿ããšã確çåäžã«è²¢ç®ãã a ãšè²¢ç®ããªã a ã®å€å¥ã¯ãéåžžã®èªç¶æ°ã®äžçã§åææ°ã ããããšããåçŽãªè©±ã§ã¯ãªãæ°ãããŠããŸãã
No.1119DD++2023幎5æ21æ¥ 16:18
ãããããããDD++ ããã
ããã©ãŒã©ãã³ã®åºã«ã¯çŽ æ°ããšãã®ãè¯ãã®ã§ããããïŒã確çã®è©äŸ¡ã«åœ±é¿ãããããŸããïŒããšããç§ã®çåã«ãçããã ãã£ãŠããããšãããããŸãã
ç§ã®ã»ãã§ã調ã¹ãŠã¿ãããšããããŸãã
Miller-Rabin çŽ æ°å€å®æ³ã¯ç¢ºççã«çŽ æ°ãããæ°ãã©ãããå€å®ããã®ã«æçšã§ããã
åºã®ãšãããã«ãã£ãŠã¯ãéå®çãªç¯å²å
ã«ãŠã確ç100ïŒ
ã§å€å®ã§ããå ŽåããããŸãã泚ç®ç¹ãšããŠã¯ããã®åºã®åãæ¹ã§ãã
â
a = {2, 3, 5, 7, 11, 13, 17} ãè©Šãã° 341550071728321 以äžã®çŽ æ°å€å®ã確ç100ïŒ
ã§ã§ããã
â¡
a = {2, 3, 5, 7, 11, 13, 17, 19, 23} ãè©Šãã° 3825123056546413051 以äžã®çŽ æ°å€å®ã確ç100%ã§ã§ãã
ããã®â ãâ¡ã§ã¯ãåºãšããŠçŽ æ°ã®ã¿ã®çµã䜿ã£ãŠããŸãã
ãªããâ ãâ¡ã¯OEISã® https://oeis.org/A014233 ã«ãããŸãã
äžå®ã®ç¯å²ã®æ°ã«ã€ããŠçŽ æ°å€å®ã確ç100%ã§å€å®ããã«ã¯
åºãšããŠã¯çŽ æ°ã®çµã䜿ãã®ãããã®ããš
æã£ããšããã§ãããäžæ¹ã«ãããŠãããã«åãããããªã
次ã®ãããªãµã€ãããããŸããã
Deterministic variants of the Miller-Rabin primality test
http://miller-rabin.appspot.com/
äžã®ãµã€ãã«ããã°7ã€ã®åºã䜿ã£ãŠ
2^64 ãŸã§ã®æ°ã確å®çã«çŽ æ°å€å®ã§ããåºã®äŸãšããŠ
2, 325, 9375, 28178, 450775, 9780504, 1795265022
ãæããããŠããŸãã
å¶æ°ã5ã®åæ°ãããåºã«ãªã£ãŠããŠ
åç¶ãšããŸããã
æ©ã¿ã¯ç¶ããŸãã
No.1127Dengan kesaktian Indukmu2023幎5æ24æ¥ 17:27
å¥çŽ æ°Pãããããå°ããçŽ æ°pãšèªç¶æ°kãçšããŠ
P=p+k*(k+1)/2
ã®åœ¢åŒã§è¡šãããã®ãæ¢ããš
3=2+1*2/2
5=2+2*3/2

13=3+4*5/2
(7+3*4/2ãå¯èœ)

31=3+7*8/2

97=19+12*13/2
(31+11*12/2,61+8*9/2ãå¯èœ)

ã®æ§ã«P,pã®éã§k*(k+1)/2ã®ç¹ãããæ§æãããŠããã
ãšããã§ããããé 匵ã£ãŠããã®æ§æãèŠã€ãããªãå¥çŽ æ°PãååšããŠããŸãã
ããã¯äœã§ããããïŒ
No.1120GAI2023幎5æ22æ¥ 07:01
7ãš61ããªïŒ
(è¿œèš)
ããã°ã©ã ãäœã£ãŠèª¿ã¹ããã211ã該åœããŠããŸããã
No.1121ãããã2023幎5æ22æ¥ 07:43
GAIæ§ãããããæ§ãããã«ã¡ã¯ã
ïŒP=p+k*(k+1)/2
ãšãããš
P=p+(1+2+3+ã»ã»ã»+k)
ãšããããšã§ããïŒ
No.1122ããããã¯ã¡ã¹ã2023幎5æ22æ¥ 12:15
ãã®3å以å€ã«èŠåœãããªãã®ãé¢çœãã£ãã§ãã
Pã倧ãããªã£ãŠãããšãæ§æã§ããå Žåã®å¯èœæ§ãå§åçã«å¢å€§ããŠããã®ã§ãã以äžã¯èŠã€ãããªãã®ã§ããããã
No.1123GAI2023幎5æ22æ¥ 12:30
ãã©ãŒã»ã©ãã³æ³ã®Storong pseudoprimeã®çºèŠã«ãããŠ
åºã2,3,6ã®ïŒã€ã§èª¿æ»ããã°1ïœ10^8ãŸã§ã«ã¯
次ã®21åãçºèŠã§ã
1373653;[829, 1; 1657, 1]
1530787;[619, 1; 2473, 1]
1987021;[997, 1; 1993, 1]
2284453;[1069, 1; 2137, 1]
3116107;[883, 1; 3529, 1]
5173601;[929, 1; 5569, 1]
6787327;[1303, 1; 5209, 1]
11541307;[1699, 1; 6793, 1]
13694761;[2617, 1; 5233, 1]
15978007;[1999, 1; 7993, 1]
16070429;[1637, 1; 9817, 1]
16879501;[1453, 1; 11617, 1]
25326001;[2251, 1; 11251, 1]
27509653;[3709, 1; 7417, 1]
27664033;[3037, 1; 9109, 1]
28527049;[2389, 1; 11941, 1]
54029741;[1733, 1; 31177, 1]
61832377;[3517, 1; 17581, 1]
66096253;[5749, 1; 11497, 1]
74927161;[6121, 1; 12241, 1]
80375707;[4483, 1; 17929, 1]
åããåºã2,3,7ã§èª¿æ»ãããšæ¬¡ã®1åã
2284453;[1069, 1; 2137, 1]
ãŸãåºã2,3,11ã§ã
2284453ããçºèŠãããã
åºã2,3,13ãªã
6787327;[1301,1;5209,1]ãåºçŸ
ãšããã§P=p+k*(k+1)/2
ãšæ§æã§ããªãã£ãã¿ã€ã(2ãå«ããŠ),2,7,61ãåºã«ããŠèª¿æ»ããã°ã
äžã€ãæ¬çŽ æ°ã¯ååšããªãããšãå€æããã
ãŠã£ãããã£ã¢ã®ãã©ãŒ-ã©ãã³çŽ æ°å€å®æ³ã®èšäºã«ããã°
ãã
n<4759123141ãªããåºã2,7,61ã«ã€ããŠèª¿ã¹ãã°ãããã
ãšããã
n=10^8ïœ10^10ã§ã¯
4759123141;[48781, 1; 97561, 1]ããåºçŸããŠããŸãã
ãããšäœãããé¢ä¿ããŠãããããšæãããŸããã
No.1125GAI2023幎5æ23æ¥ 05:43
ä»»æã®æ£ã®æŽæ° n ã«ã€ããŠ
(334*10^n -1)/9
ã¯ãåžžã«åææ°ã§ãã
ããŠãã©ãããŠã§ããããïŒ
No.1080Dengan kesaktian Indukmu2023幎5æ13æ¥ 22:03
æ£ã®æŽæ° n ã«ã€ããŠ
(109*10^n -1)/9
ã¯ãããªããããåææ°ãšã¯ããããŸããã
ããŠãã©ãããŠã§ããããïŒ
No.1081Dengan kesaktian Indukmu2023幎5æ13æ¥ 22:19
[åè
]
ååã3339,33399,333999,3339999,âŠã®ããã«ãªã
n=3mã®ãšã
ååã333999,333999999,333999999999,âŠã®ããã«3m+3æ¡ã ãã
333ã§å²ãåãã333÷9=37ãªã®ã§äžåŒã¯37ã§å²ãåãã
n=3m-1ã®ãšã
ååã¯33399,33399999,33399999999,âŠã ãã
äžåŒã¯3711,3711111,3711111111,âŠã®ããã«ãªã
å
šæ¡ã®åèšã3ã®åæ°ã ãã3ã§å²ãåãã
n=6m-2ã®ãšã
ååã¯3339999,3339999999999,3339999999999999999,âŠã®ããã«ãªã
3339999÷13=256923ã999999÷13=76923ã ãã
åå÷13ã¯256923,256923076923,256923076923076923,âŠãšãªãååã¯13ã§å²ãåãã
ãããŠ13ã¯åæ¯ã®9ãšäºãã«çŽ ã ããäžåŒã13ã§å²ãåãã
n=6m-5ã®ãšã
ååã¯3339,3339999999,3339999999999999,âŠã®ããã«ãªã
3339÷7=477ã999999÷7=142857ã ãã
åå÷7ã¯477,477142857,477142857142857,âŠãšãªãååã¯7ã§å²ãåãã
ãããŠ7ã¯åæ¯ã®9ãšäºãã«çŽ ã ããäžåŒã7ã§å²ãåãã
ãŸãšã
n=3mã®ãšã37ã§å²ãåãã
n=3m-1ã®ãšã3ã§å²ãåãã
n=6m-2ã®ãšã13ã§å²ãåãã
n=6m-5ã®ãšã7ã§å²ãåãããã
åžžã«åææ°ã
[åŸè
]
n=136ã®ãšãçŽ æ°ãšããåäŸãããããåææ°ãšã¯éããªãã
No.1082ãããã2023幎5æ14æ¥ 02:59
çŽ æŽãããã§ãã
ãªããã¿å
ã¯ä»¥äžã§ãã
https://oeis.org/A112386
远䌞ïŒ
https://oeis.org/A107612
ãã¡ãããåèã«ããŸããã
No.1083Dengan kesaktian Indukmu2023幎5æ14æ¥ 10:23
A112386ã¯ãªããããªã«ã¡ãã£ãšããããŒã¿ããªããã§ããããã
ãã£ãã37ã®äŸãèŒããŠãããã ãã
251,26111,271,281,29111111,3011,311,3211111111111111111111111111111111111,34111111,3511,361111,-1
ãããèŒããã°ããã®ã«ã
ïŒå
é ã®3è¡ã®ãšããã§ã¯ãªãLINKSã«ããè¡šã®ããšã§ãïŒ
ã¡ãã£ãšèª¿ã¹ãŠããããè¿œå ããŠã¿ããããªã
No.1084ãããã2023幎5æ14æ¥ 13:03
æ£æ»æ³ã§çé¢ç®ã«çŽ å æ°å解ãããŠãããš
56111âŠ11
(505*10^n -1)/9
ã§ã®çŽ æ°æ¢ãã§èšç®éãççºããã®ã§ã¯ãš
å±æ§ããããŸãã
No.1086Dengan kesaktian Indukmu2023幎5æ14æ¥ 13:48
以åãã£ã 381111âŠâŠ ã®è©±ããšæã£ãããä»å㯠371111âŠâŠ ãªã®ã§ããã
http://shochandas.xsrv.jp/mathbun/mathbun1315.html
No.1087DD++2023幎5æ14æ¥ 15:39
381âŠ1
ã¯å人çã¯æªè§£æ±ºã ã£ãã®ã§ããã
ããã¯
(343*10^n -1)/9
ã§ãã
ïŒ1ïŒn â¡ 0 (mod 3)
ã®ãšãã«ã¯ã
n = 3*k
ãšãããŠ
343 = 7^3
ã«çæãããš
((7*10^k)^3 -1^3)/9
ã§ããããååã¯ïŒ3ä¹â3ä¹ïŒã®åœ¢ãšãªã
å æ°å解ã§ããããšãã
åææ°ã§ããããã§ãã
ïŒïŒïŒn â¡ 1 (mod 3)
ã®ãšãã«ã¯
381, 381111, 381111111âŠâŠ
ãªã©ã§ããã
ïŒã®åæ°ã§ãã
ïŒïŒïŒn â¡ 2 (mod 3)
ã®ãšãã«ã¯
ã©ããããã®ã§ããããïŒ
No.1089Dengan kesaktian Indukmu2023幎5æ14æ¥ 18:08
ãã£ããã§ããã
ïŒïŒïŒn â¡ 2 (mod 3)
ãšããããšã¯
3811, 3811111, 3811111111
ã§ããã
3811 ã¯ã37 ã®åæ°ã
111 ã¯ã37 ã®åæ°ã§ããã
No.1090Dengan kesaktian Indukmu2023幎5æ14æ¥ 18:13
(a10^n-1)/9ãšãa10^n-1ã«ãããã©ãã§ãããã
aã¯2m,2m+1ãšãã3m,3m+1,3m+2ãšãã10m,10m+1,10m+2,10m+3,10m+4,10m+5,10m+6,10m+7,10m+8,10m+9ãªããã©ãã§ãããã»ã»ã»ã»ã»
No.1091ããããã¯ã¡ã¹ã2023幎5æ15æ¥ 07:12
äžå¿èšãåºãã£ãºãªã®ã§ã圢ã ãã§ããªããšãã»ã»ã»ã»ã»
ïŒïŒa=10mãšãããšã(mã¯èªç¶æ°ïŒ
a10^n-1=10m10^n-1=(m-1)10^(n+1)+10^(n+1)-1
ããã§ã10^(n+1)-1=99ã»ã»99=(11ã»ã»11)x9
11ã»ã»11ãèãããš1ãn+1䞊ãã æ°ãªã®ã§ãn+1ãåææ°ãªããããªããå²ããã
n+1=6=2x3ãªã®ã§ã111111=11x(10101)=111x(1001)=11x3x7x13x37=(111)x(1001)=(3x37)x(7x11x13)
ãããããn+1ãå¶æ°ãªãã11ã®åæ°ã
n+1ã3ã®åæ°ãªãã111(=3x37)ã®åæ°ã
n+1ã5ã®åæ°ãªãã11111(=41x271)ã®åæ°ã
n+1ã7ã®åæ°ãªãã1111111(=239x4649)ã®åæ°ã
n+1ã11ã®åæ°ãªãã11111111111(=21649x513239)ã®åæ°ã
n+1ã13ã®åæ°ãªãã1111111111111(=53x79x265371653)ã®åæ°ã
以äžçç¥
ãŸãã(m-1)10^(n+1)ã¯ã(m-1)x2^(n+1)x5^(n+1)
ãããã£ãŠã
n+1ãå¶æ°ãªããm-1ã¯11ã®åæ°ãªããa10^n-1ã¯11ã®åæ°ã
n+1ã3ã®åæ°ãªããm-1ã¯3ã37ã®åæ°ãªããa10^n-1ã¯3ã37ã®åæ°ã
n+1ã5ã®åæ°ãªããm-1ã¯41ã271ã®åæ°ãªããa10^n-1ã¯41ã271ã®åæ°ã
以äžçç¥
ïŒïŒa=10m+1ãšãããšã(mã¯èªç¶æ°ïŒ
a10^n-1=(10m+1)10^n-1=m10^(n+1)+10^n-1
ããã§ã10^n-1=99ã»ã»99=(11ã»ã»11)x9
11ã»ã»11ãèãããš1ãn䞊ãã æ°ãªã®ã§ãnãåææ°ãªããããªããå²ããã
n=6=2x3ãªã®ã§ã111111=11x(10101)=111x(1001)=11x3x7x13x37=(111)x(1001)=(3x37)x(7x11x13)
ãããããnãå¶æ°ãªãã11ã®åæ°ã
nã3ã®åæ°ãªãã111(=3x37)ã®åæ°ã
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nã11ã®åæ°ãªãã11111111111(=21649x513239)ã®åæ°ã
nã13ã®åæ°ãªãã1111111111111(=53x79x265371653)ã®åæ°ã
以äžçç¥
ãŸããm10^(n+1)ã¯ãmx2^(n+1)x5^(n+1)
ãããã£ãŠã
nãå¶æ°ãªããmã¯11ã®åæ°ãªããa10^n-1ã¯11ã®åæ°ã
nã3ã®åæ°ãªããmã¯3ã37ã®åæ°ãªããa10^n-1ã¯3ã37ã®åæ°ã
nã5ã®åæ°ãªããmã¯41ã271ã®åæ°ãªããa10^n-1ã¯41ã271ã®åæ°ã
以äžçç¥
ïŒïŒa=10m+2ãšãããšã(mã¯èªç¶æ°ïŒ
a10^n-1=(10m+2)10^n-1=(m+1)10^(n+1)+10^n-1
ããã§ã10^n-1=99ã»ã»99=(11ã»ã»11)x9
11ã»ã»11ãèãããš1ãn䞊ãã æ°ãªã®ã§ãnãåææ°ãªããããªããå²ããã
n=6=2x3ãªã®ã§ã111111=11x(10101)=111x(1001)=11x3x7x13x37=(111)x(1001)=(3x37)x(7x11x13)
ãããããnãå¶æ°ãªãã11ã®åæ°ã
nã3ã®åæ°ãªãã111(=3x37)ã®åæ°ã
nã5ã®åæ°ãªãã11111(=41x271)ã®åæ°ã
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No.1098ããããã¯ã¡ã¹ã2023幎5æ16æ¥ 11:04
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æ£ããã¯æš¡ç¯è§£çã® PDF ã«ãããŸããšããã
If n = 3*k â¡ 0 (mod 3), observe that
9*a*(3*k) = 34299 · · · 99
= (7*10^k)^3 â1
which is properly divisible by 7*10^k â 1,
a number that is larger than 9.
Hence a(3k) admits a non-trivial factor and so is not prime.
ãªã®ã§ãããç§ã¯ãšãã§ããªãèªã¿ééãã
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If n = 3*k â¡ 0 (mod 3), observe that
9*a*(3*k) = 34299 · · · 99
= (7*10^k)^3 â1
which is properly divisible by (7*10^k)^3 â1
,
a number that is larger than 9.
Hence a(3k) admits a non-trivial factor and so is not prime.ã
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ãããŠæ®ã£ã No.1051 ã®æçåŒã®è¬ã
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No.1056DD++2023幎5æ6æ¥ 11:45
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nCk ã C[n,k] ãšæžãããšã«ããŸãã
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C[n,i] * C[n-i,j] = C[n,j] * C[n-j,i]
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f(k)ãkã®n-1次以äžã®å€é
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Σ[k=0...n] C[n,k] * (-1)^k * f(k) = 0
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Σ[h=0...n] C[n,h] * (h+1)^h * (n-h+1)^(n-h-1)
= Σ[h=0...n] C[n,n-h] * (n-h+1)^(n-h-1) * (h+1)^h
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * (n-k+1)^(n-k)
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * ((n+2)-(k+1))^(n-k)
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * { Σ[m=0...n-k] C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-k-m) }
= Σ[k=0...n] Σ[m=0...n-k] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= Σ[m=0...n] Σ[k=0...n-m] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] Σ[k=0...n-m] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] Σ[k=0...n-m] C[n,m] * C[n-m,k] * (n+2)^m * (-1)^(n-m) * (-1)^k * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] C[n,m] * (n+2)^m * (-1)^(n-m) * { Σ[k=0...n-m] C[n-m,k] * (-1)^k * (k+1)^(n-m-1) }
= (n+2)^n + Σ[m=0...n-1] C[n,m] * (n+2)^m * (-1)^(n-m) * 0
= (n+2)^n
No.1068ããã²ã2023幎5æ8æ¥ 19:12
ããããªãã»ã©ããããªæ¹æ³ã§ k+1 ã®ææ°ãã k ãæ¶ãããšã¯ã
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> Σ[k=0...n] nCk * (k+1)^k * (n-k+1)^(n-k-1) = (n+2)^n
>
> Σ[k=0...n] nCk * (k+1)^(k-1) * (n-k+1)^(n-k-1) = 2*(n+2)^(n-1)
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