ãã§ã«ããŒã®æçµå®çã«ææŠãçŽãã§ãã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
ããã i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
ããã i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
ãããi=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1=c^n-1
(a^n-1)+(b^n-1)=(c^n-1)
(a^n-1)=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ãããn
a^n-1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãããi=1
ããã§ã(a^n-1)=(c^n-1)-(b^n-1)ãæãç«ã€ã«ã¯ã
ïŒïŒi=nã®ãšããa^n-1ã®nCné
ã¯ã
nCn{1^0+2^0+3^0+ã»ã»ã»+(a-1)^0)}=nCn{a-1}=a-1----(d)
äžæ¹(c^n-1)-(b^n-1)ã®nCné
ã¯ã
nCn{b^0+(b+1^0+(b+2)^0ã»ã»ã»+(c-1)^0}=nCn{(c-1)-(b-1)}=c-b---(e)
åŒ(d),(e)ãçå·ã§çµã°ããã®ã¯ã
c-b=a-1---(i)
ã®ãšãã ãã§ããã
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
äžæ¹(c^n-1)-(b^n-1)ã®nC(n-1)é
ã¯ã
nC(n-1){b+(b+1)+(b+2)ã»ã»ã»+(c-1)}=n{(c-1)c/2-(b-1)b/2}---(g)
åŒ(f),(g)ãçå·ã§çµã°ããã®ã¯ã
(c-1)c/2-(b-1)b/2=(a-1)a/2
ã®ãšãã ãã§ããã
(c-1)c-(b-1)b=(a-1)a
c^2-c-b^2+b=(a-1)a
c^2-b^2-(c-b)=(a-1)a
(c-b)(c+b-1)=(a-1)a
åŒ(d),(e)ãçãããšãåŒ(f),(g)ãçãããªããšãããªããããåŒ(i)ããã
c+b-1=aã巊蟺ã(c-b)ã§å²ã£ãŠãå³èŸºã(a-1)ã§å²ã£ãŠïœãªããªãåŒ(i)ããïœ
c+b-2=a-1=c-bãåŒ(i)ãã
c+b-2=c-b
c+b-2-(c-b)=0
c+b-2-c+b=0
2b-2=0
b=1
ããã¯ãc>b>aã«ççŸããã
ãããã£ãŠã
(d)â (e)ã(f)â (g)
ã€ãŸãã
(a^n-1)â (c^n-1)-(b^n-1)
a^nâ c^n-b^n
a^n+b^nâ c^n
ãã£ãŠããã§ã«ããŒã®æçµå®çã¯åççã«èšŒæãããã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
äºé
å®çã®ææžã®åŒçšã¯ç·è²ã®ãããããã¯ã¡ã¹ãããã¯ãªãã¯ããã°ãéããŸãã
> a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
> a^n-1+b^n-1=c^n-1
巊蟺㧠2 å -1 ãããªããå³èŸºã 2 å -1 ããå¿
èŠãããã®ã§ã¯ã
No.872DD++4æ11æ¥ 16:16
äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
i=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1+1=c^n-1
(a^n-1)+(b^n-1)+1=(c^n-1)
(a^n-1)+1=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
ããã§ã(a^n-1)+1=(c^n-1)-(b^n-1)ãæãç«ã€ã«ã¯ã
ïŒïŒi=nã®ãšããa^n-1ã®nCné
ã¯ã
nCn{1^0+2^0+3^0+ã»ã»ã»+(a-1)^0)}+1=nCn{a-1}+1=a----(d)
äžæ¹(c^n-1)-(b^n-1)ã®nCné
ã¯ã
nCn{b^0+(b+1^0+(b+2)^0ã»ã»ã»+(c-1)^0}=nCn{(c-1)-(b-1)}=c-b---(e)
åŒ(d),(e)ãçå·ã§çµã°ããã®ã¯ã
c-b=a---(i)
ã®ãšãã ãã§ããã
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
äžæ¹(c^n-1)-(b^n-1)ã®nC(n-1)é
ã¯ã
nC(n-1){b+(b+1)+(b+2)ã»ã»ã»+(c-1)}=n{(c-1)c/2-(b-1)b/2}---(g)
åŒ(f),(g)ãçå·ã§çµã°ããã®ã¯ã
(c-1)c/2-(b-1)b/2=(a-1)a/2
ã®ãšãã ãã§ããã
(c-1)c-(b-1)b=(a-1)a
c^2-c-b^2+b=(a-1)a
c^2-b^2-(c-b)=(a-1)a
(c-b)(c+b-1)=(a-1)a
åŒ(d),(e)ãçãããšãåŒ(f),(g)ãçãããªããšãããªããããåŒ(i)ããã
c+b-1=a-1ã巊蟺ã(c-b)ã§å²ã£ãŠãå³èŸºãaã§å²ã£ãŠïœãªããªãåŒ(i)ããïœ
c+b=a=c-bãåŒ(i)ãã
c+b=c-b
c+b-(c-b)=0
c+b-c+b=0
2b=0
b=0
ããã¯ãc>b>aã«ççŸããã
ãããã£ãŠã
(d)â (e)ã(f)â (g)
ã€ãŸãã
(a^n-1)â (c^n-1)-(b^n-1)
a^nâ c^n-b^n
a^n+b^nâ c^n
ãã£ãŠããã§ã«ããŒã®æçµå®çã¯åççã«èšŒæãããã
(d) ãš (e) ãçãããšãããæ ¹æ ã¯ãªãã§ããïŒ
No.876DD++4æ11æ¥ 17:51
(a^n-1)+1=(c^n-1)-(b^n-1)ãæãç«ã€ããã§ãã
ã€ãŸãã
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
ãšããããšã§ãå³èŸºã¯ãã¹ãŠæ£ã®æ°ãªã®åãªã®ã§ãã
ãŸããa^n-1+1ã¯
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ããã¹ãŠãæ£ã®æ°ã®åã§ãããã
nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}
ã®å³å·ŠèŸºã®nCiã©ããçãããªããã°ãªããŸããã
åŒ(d),(e)ã¯ãnCnã®é
ãªã®ã§ã
a^n-1+1ã¯i=nã®
nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
ã§ããããã®é
ã ããïŒïŒãäœåã«ããã(c^n-1)-(b^n-1)ã¯ãi=nã®
nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ã§ãããããäºãã«çãããªããã°ãªããŸãããã ããåŒ(d),(e)ã¯ãçãããªããã°ãªããŸããã
äºé
å®çã§ãåãã¹ãä¹ãªãã
(a+b)^nã®åé
ã¯ãnCi a^(n-i) b^iã§ã(a+b)^n=(c+d)^nãªãã a^(n-i) b^i= c^(n-i) d^iãšããããšã§ãã
ã€ãŸããnCiã®ä¿æ°é
ã¯a^(n-i) b^i= c^(n-i) d^iã®ããã«çãããªããªããã°ãªããªããšããããšã§ãã
No.877ããããã¯ã¡ã¹ã4æ11æ¥ 19:36
ïŒïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}+1=nC(n-1){(a-1)a/2}----(f)
ãããªãã§ããããã
No.878KY4æ11æ¥ 20:45
> (a+b)^n=(c+d)^nãªãã a^(n-i) b^i= c^(n-i) d^iãšããããšã§ãã
a=1, b=-1, c=0, d=0 ã§èãããšã
ã(1-1)^n = (0+0)^n ãªãã1^(n-i) (-1)^i = 0^(n-i) 0^i ãšããããšãã£ãŠæå³ã«ãªããŸããã©ããã£ãŠãŸãïŒ
No.879DD++4æ11æ¥ 21:58
KYæ§ããã¯ããããããŸãã
nC(n-1)=n!/(n-(n-1)!(n-1)!)=n!/(n-1)!=n
ãªã®ã§ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}=n{(a-1)a/2}----(f)
ãŸãã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}+1=nC(n-1){(a-1)a/2}---(f)
ïŒïŒã¯ãnCnã®é
ã ãã«äœçšããŸãã®ã§ãnC(n-1)ã«ã¯ãé¢ä¿ããŸããã
ã§ãããã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}=n{(a-1)a/2}----(f)
ã§ããã¯ãã§ãã
DD++æ§ããã¯ããããããŸãã
ãããããµãã«ããã°ããããªããŸããã
æçš¿å¶éãããã£ãŠããã®ã§ãããã«æžããŸãã
ïŒãªãã»ã©ãã€ãŸããã¯ã¡ã¹ããã㯠(-1)^i = 0^n ãæ£ãããšåºåŒµããŠããããã§ããïŒ
ä»åã®å Žåãa,b,c,dãšãã«ãèªç¶æ°ã§ããããããã¯ãªããªããšæããŸãã
ãææã®ã
ïŒã(1-1)^n = (0+0)^n ãªãã1^(n-i) (-1)^i = 0^(n-i) 0^i ãšããããšã
ã§ãããã(1-1)^n=0ã(0+0)^n=0ã§ãå
šäœã§èŠãã°ãçå·ãæãç«ã¡ãŸããã1^(n-i) (-1)^i = 0^(n-i) 0^iãšã¯ãèšããªãã§ããã
ã¡ãªã¿ã«ã(1-1)^nã¯ã
(1-1)^n=nC0 1^n (-1)^0+nC1 1^(n-1) (-1)^1+nC2 1^(n-2) (-1)^2+nC3 1^(n-3) (-)1^3+ã»ã»ã»ã»+nC(n-1) 1^(n-(n-1)) (-1)^(n-1)+nCn 1^(n-n) (-1)^n
ã«ãããŠã
nãå¶æ°ãªããããšãã°n=10ãªãã
0=10C0-10C1+10C2-10C3+10C4-10C5+10C6-10C7+10C8-10C9+10C10
ãã€ãã¹ã®é
ã巊蟺ã«ç§»é
ãããšã
10C1+10C3+10C5+10C7+10C9=10C0+10C2+10C4+10C6+10C8+10C10
ãã£ãŠã
nC1+nC3+nC5ã»ã»ã»+nC(n-1)=nC0+nC2+nC4+ã»ã»ã»ã»+nCn
å·Šå³ã§é
æ°ãéãã®ã«äžæè°ã«æããããããŸãããããããªã®ã§ãã
nã奿°ãªããããšãã°n=11ãªãã
0=11C0-11C1+11C2-11C3+11C4-11C5+11C6-11C7+11C8-11C9+11C10-11C11
ãã€ãã¹ã®é
ã巊蟺ã«ç§»é
ãããšã
11C1+11C3+11C5+11C7+11C9+11C11=11C0+11C2+11C4+11C6+11C8+11C10
ãã£ãŠã
nC1+nC3+nC5ã»ã»ã»+nCn=nC0+nC2+nC4+ã»ã»ã»ã»+nC(n-1)
ãã¹ã«ã«ã®äžè§åœ¢ãæãåºããŠãã ããã
ããããããããã1ã-2ã1
ãããããããã1ã-3ã3ã-1
ããããããã1ã-4ã6ã-4ãã1
ãããããã1ã-5ã10ã-10ã5ã-1
ããããã1ã-6ã15ã-20ã15ã-6ã1
ãšãªããŸãã
No.880ããããã¯ã¡ã¹ã4æ12æ¥ 07:20
ãªãã»ã©ãã€ãŸããã¯ã¡ã¹ããã㯠(-1)^i = 0^n ãæ£ãããšäž»åŒµããŠããããã§ããïŒ
No.881DD++4æ12æ¥ 07:24
æŽæ°èšäºãèŠãŠãæ²ç€ºæ¿ã«ãªãã¯ãã®è¬ã®è¿ä¿¡ãæ¥ãŠãããšæã£ããâŠâŠã
è¿äºã¯å¿
ãæ°ããã¡ãã»ãŒãžã§æžããŠãã ããã
éå»ã®æçš¿ã«å çããŠè¿äºããããŠãæ°ã¥ããŸããã
ãªãã»ã©ãèªç¶æ°éå®ã ãããšãã£ããããªãããããŸãããã
a = 1, b = 3, c = 2, d = 2 ã§èããŸãã
æå¥ãªãèªç¶æ°ã§ããïŒ
ã§ã(1+3)^n = (2+2)^n ã¯æãç«ã¡ãŸãããããåé¡ãªãã§ããïŒ
ãšããããšã¯ãã¯ã¡ã¹ããã㯠1^(n-i) 3^i = 2^(n-i) 2^i ã§ããããšã
ã€ãŸã 3^i = 2^n ã¯æ£ããåŒã§ãããšäž»åŒµããããã§ããïŒ
ã¯ã¡ã¹ãããããã®åŒã誀ãã ãšæãããªãããŸã£ããåãè«çã§äœã£ã (d) = (e) ã誀ããšããããšã§ãã
No.882DD++4æ12æ¥ 16:01
DD++æ§ããã¯ããããããŸãã
ãã®ãšããã§ããã
ãã®ãã§ã«ããŒã®æçµå®çã®èšŒæã¯ãééãã§ããã
No.883ããããã¯ã¡ã¹ã4æ13æ¥ 06:59
äŒãã£ãããã§ãããã£ãã§ãã
No.884DD++4æ13æ¥ 07:48
äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
i=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1+1=c^n-1
(a^n-1)+(b^n-1)+1=(c^n-1)
(a^n-1)+1=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
n
Σ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}-(a)åŒ-1----(d)
i=1
ãšãããšã(d)åŒã®
(c-1)^(n-i)-(a-1)^(n-i)
ã®å€§å°é¢ä¿ã調ã¹ãã°ããã
å
Œ΋
x^n-y^n=(x-y){x^(n-1)+x^(n-2)y+x^(n-3)y^2+ã»ã»ã»+xy^(n-2)+y^(n-1)}
ããã
x,yãèªç¶æ°ãªãã{}ã®äžã¯ãæ£ã®èªç¶æ°ããããã£ãŠã(x-y)ãæ£ãè² ã§x^nãšy^nã®å€§å°é¢ä¿ããããã
(c-1)^(n-i)-(a-1)^(n-i)
ã«ãããŠãc>b>aãããc-1>a-1ããã
(c-1)^(n-i)-(a-1)^(n-i)>0
ãšãªãããã£ãŠ(d)åŒã¯>0
ãã ãc-bã®é
æ°ãšaã®é
æ°ãåé¡ãšãªãã
ãããã£ãŠãæ¡ä»¶ã¯c-bâ§aãã€ãã
ãããæºè¶³ããã°ããã§ã«ããŒã®æçµå®çã¯èšŒæã§ããã
ãªãã(a)åŒ+1ã®éšåã¯ãb^0-1-1>0ã¯a,b,cã¯èªç¶æ°ã§ãããc>b>a>0ãša=1ã§ã¯ãb>3ã§ããããåé¡ãªãã
ããšãã°ãa=1ã®ãšãã1^3+b^3=c^3ã®ãšãb=2ã§ã3ã§ã¯ãªãã
No.898ããããã¯ã¡ã¹ã4æ15æ¥ 07:08
a,b,cã«ãããŠã
a^n+b^n=c^n
ãæãç«ã€ãšãã
(a^n+b^n)^2=(c^n)^2
ããã§ã
http://y-daisan.private.coocan.jp/html/pdf/felmer-5-4.pdfïŒç·è²ã®ããããã¯ã¡ã¹ããã¯ãªãã¯ããã°éããŸããïŒ
ã®è£é¡ããã
(a^n+b^n)^2>(a+b)^n
ã§ããããã
(a^n+b^n)^2=(c^n)^2
(a+b)^n<(c^2)^n
a,b,cã¯èªç¶æ°ããã
(a+b)<c^2
a<c^2-b
ããããªããc^2-b>aãªããå¶éããªããªã£ãã®ã«ãªãã
ïŒããããªããc^2-b>aãªããå¶éããªããªã£ãã®ã«ãªãã
ããã¯ã©ãããäºãæå³ããŠããã®ã§ãããããå¶éãä»ããŠè¡ã£ãŠããåŸãªã蚌æãããã®ãçãªã®ã§ã¯ãªãã§ããããã
å ã¿ã«ãïœïŒïœïŒïœïŒïœããïœïŒïœïŒïŒïœã§ãããïœâ§ïŒã§ïœïŒïœïŒïœ^2ããïœïŒïœã«å¶éãä»ããããŸããã
è£é¡ã®èšŒæã¯èŠäºã§ããã
No.909KY4æ15æ¥ 13:21
KYæ§ãããã«ã¡ã¯ã
ä»ç§ã¯ã24æéã§20ä»¶ã®æçš¿å¶éã§ãäœãæ¶ããªããšæçš¿ã§ããªãã®ã§ããç¡çãã1ã€æ¶ããŸããã
ïŒå ã¿ã«ãïœïŒïœïŒïœïŒïœããïœïŒïœïŒïŒïœã§ãããïœâ§ïŒã§ïœïŒïœïŒïœ^2ããïœïŒïœã«å¶éãä»ããããŸããã
ãªãã»ã©ãããšã¡ãã£ãšã§ã»ã»ã»ã»ã»
å¶éãªãã«ãªãã°ããã§ã«ããŒã®æçµå®çã®åçç蚌æã«ãªã£ããã§ããã©ãã
æ®å¿µã
No.910ããããã¯ã¡ã¹ã4æ15æ¥ 16:23
> è£é¡ã®èšŒæ
aâ§2, bâ§2 ã®ãšããè«ç¹å
åã§äžçºéå Žã§ã¯ã
å
¥è©Šãšãã ãšäžè¡èªãã ã ãã§ 0 ç¹ã«ããããã€ã§ãã
No.912DD++4æ15æ¥ 17:46
ããããc-b<aã®ãšãã(d)åŒã¯<0ã§ãã
èŠããã«ã(d)åŒã=0ã§ãªããã°ããã§ã«ããŒã®æçµå®çã®åçç蚌æã¯ã§ãããã ã
ãªããšããå
ãèŠããŠããŸããã
DD++æ§ã®ææã®
ïŒã§ã(1+3)^n = (2+2)^n ã¯æãç«ã¡ãŸãããããåé¡ãªãã§ããïŒ
ãšããããšã¯ãã¯ã¡ã¹ããã㯠1^(n-i) 3^i = 2^(n-i) 2^i ã§ããããšã
ã€ãŸã 3^i = 2^n ã¯æ£ããåŒã§ãããšäž»åŒµããããã§ããïŒ
ããããå®ã«ãããããææã§ãa^n-1+1=c^n-1-(b^n-1)ããæãç«ã€æ¡ä»¶ã¯ãªããšããããæå³ãããã³ããªããããããªãã»ã»ã»ã»ã»
No.913ããããã¯ã¡ã¹ã4æ15æ¥ 18:59
ïŒaâ§2, bâ§2 ã®ãšããè«ç¹å
åã§äžçºéå Žã§ã¯ã
ããããè«ç¹å
åã§ã¯ãããŸããããŸããæ¬é¡ã®èšŒæã®æ¹ã§ïœïŒïŒïŒïœïŒïŒã®å Žåãè¿°ã¹ãŠããŠã次ã«è£é¡ã®æã§ïœïŒïŒïŒïœïŒïŒïŒïœïŒïŒïŒïœïŒïŒïŒã®å Žåãè¿°ã¹ãŠããŠãæ®ãã¯ïœâ§ïŒïŒïœâ§ïŒã®å Žåãããªãããã§ãã
å ã¿ã«ãå
·äœäŸã¯ã
ãããã¯è«ç¹å
åã ããšèšããã®ã¯ã1ã€ã®äžæ®µè«æ³ã®äžã§ã埪ç°è«æ³ãã䜿ãããŠããå Žåã§ãããããªãã¡ãæšè«éçšã«èšŒæãã¹ãäºæãåæãšããåœé¡ãå«ãã§ããå Žåã§ãããæ¬è³ªçã«ãåœé¡ãããèªèº«ã®èšŒæã«äœ¿ããããããªæŠè¡ã¯ãã®åºæ¬ç圢åŒã«ãããŠèª¬åŸåããªããäŸãã°ãããŒã«ãæ¬åœã®ããšãèšã£ãŠãããšèšŒæããããšããã
ããŒã«ã¯åãèšã£ãŠããªããšä»®å®ããã
ããŒã«ã¯äœãã話ããŠããã
ãããã£ãŠãããŒã«ã¯æ¬åœã®ããšãèšã£ãŠããã
ãã®æç« ã¯è«ççã ãã話è
ã®ç宿§ãçŽåŸãããããšã¯ã§ããªããåé¡ã¯ãããŒã«ã®ç宿§ã蚌æããããã«ããŒã«ãæ¬åœã®ããšãèšã£ãŠãããšä»®å®ããããšãèŽè¡ã«é Œãã§ãããããããã¯å®éã«ã¯ãããŒã«ãåãã€ããŠããªããªããããŒã«ã¯çå®ãèšã£ãŠããããšããããšã蚌æããŠããã«éããªãã
ãã®ãããªè«èšŒã¯è«ççã«ã¯åŠ¥åœã§ãããããªãã¡ãçµè«ã¯å®éã«åæããå°ãåºãããŠããããã ããäœããã®æå³ã§ãã®çµè«ã¯åæãšåäžã§ãããèªå·±åŸªç°è«æ³ã¯å
šãŠããã®ãããªèšŒæãã¹ãåœé¡ãè«èšŒã®ããæç¹ã§ä»®å®ããããšããæ§è³ªãæã€ã
åŒçšå
ïŒhttps://ja.wikipedia.org/wiki/%E8%AB%96%E7%82%B9%E5%85%88%E5%8F%96#%E5%85%B7%E4%BD%93%E4%BE%8B
åœãŠã¯ãŸã£ãŠããªããšæããŸããã
No.914å£ããæ4æ15æ¥ 19:53
äŸãšã㊠1 ã€ã®äžæ®µè«æ³ãæããŠããã ãã§ãè€æ°ã®å Žåã§ãè«ç¹å
åã¯è«ç¹å
åã§ãããã
ãããã¯åŸªç°è«æ³ãšèšã£ãæ¹ãããã£ãã§ããïŒ
ä»åã®å Žåãªãã©ã£ã¡ã«ã該åœããïŒãšãããäž¡è
ã«æç¢ºãªåºåãããããã§ããªãïŒãšæã£ãŠããã®ã§ã
No.915DD++4æ15æ¥ 20:35
DD++æ§ããã¯ããããããŸãã
bâ 0ãšãããa/bã»ã»ã»
ã¯ãã©ããªããã ããïŒ
No.917ããããã¯ã¡ã¹ã4æ16æ¥ 07:33
ãããå¿
ã p åã§ãã
ã¯ã¡ã¹ãããã¯ãp = 7 ãéžãã§ããã®ã« 8 åãã£ãŠãŸããã
No.905DD++4æ15æ¥ 11:11
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ãæ±ºããŠãã ããã
p=7ãa=5ããšããŸãã
ã1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
1+2p=15
ã2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
2+3p=23
ã3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
3+3p=24
ã4, 4+p, 4+2p, âŠâŠ, 4+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
4+4p=32
ã5, 5+p, 5+2p, âŠâŠ, 5+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
5+p=12
ã6, 6+p, 6+2p, âŠâŠ, 6+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
6+p=13
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{12,13,15,23,24,32}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{12,13,15,23,24,32,35}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{13,15,23,24,32,35}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{1,3,11,12,20,23}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
{1,3,11,12,20,23}â¡{1,3,4,5,6,2}(mod p=7)
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
(1)
{1,3,11,12,20,23}
{1,3,11,12,20,23,35}
{3,11,12,20,23,35}
{2,10,11,19,22,34}
(2)
{2,10,11,19,22,34}
{2,10,11,19,22,34,35}
{10,11,19,22,34,35}
{8,9,17,20,32,33}
(3)
{8,9,17,20,32,33}
{8,9,17,20,32,33,35}
{9,17,20,32,33,35}
{1,9,12,24,25,27}
(4)
{1,9,12,24,25,27}
{1,9,12,24,25,27,35}
{9,12,24,25,27,35}
{8,11,23,24,26,34}
(5)
{8,11,23,24,26,34}
{8,11,23,24,26,34,35}
{11,23,24,26,34,35}
{3,15,16,18,26,27}
(6)
{3,15,16,18,26,27}
{3,15,16,18,26,27,35}
{15,16,18,26,27,35}
{12,13,15,23,24,32}
çã
{12,13,15,23,24,32}
No.907ããããã¯ã¡ã¹ã4æ15æ¥ 12:12
ããã§ããããã§åã£ãŠãŸãã
No.908DD++4æ15æ¥ 12:15
DD++æ§ãããã°ãã¯ã
ä»ç§ã¯ã24æéã§20ä»¶ã®æçš¿å¶éã§ãäœãæ¶ããªããšæçš¿ã§ããªãã®ã§ããç¡çãã1ã€æ¶ããŸããã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ãæ±ºããŠãã ããã
p=7ãa=5ããšããŸãã
ã1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
1+p=8
ã2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
2+2p=16
ã3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
3+3p=24
ã4, 4+p, 4+2p, âŠâŠ, 4+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
4+4p=32
ã5, 5+p, 5+2p, âŠâŠ, 5+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
5+3p=26
ã6, 6+p, 6+2p, âŠâŠ, 6+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
6+p=13
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{8,13,16,24,26,32}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{8,13,16,24,26,32,35}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{13,16,24,26,32,35}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{5,8,16,18,24,27}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
{5,8,16,18,24,27}â¡{5,1,2,4,3,6}(mod p=7)
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
(1)
{5,8,16,18,24,27}
{5,8,16,18,24,27,35}
{8,16,18,24,27,35}
{3,11,13,19,22,30}
(2)
{3,11,13,19,22,30}
{3,11,13,19,22,30,35}
{11,13,19,22,30,35}
{8,10,16,19,27,32}
(3)
{8,10,16,19,27,32}
{8,10,16,19,27,32,35}
{10,16,19,27,32,35}
{2,8,11,19,24,27}
(4)
{2,8,11,19,24,27}
{2,8,11,19,24,27,35}
{8,11,19,24,27,35}
{6,9,17,22,25,33}
(5)
{6,9,17,22,25,33}
{6,9,17,22,25,33,35}
{9,17,22,25,33,35}
{3,11,16,19,27,29}
(6)
{3,11,16,19,27,29}
{3,11,16,19,27,29,35}
{11,16,19,27,29,35}
{8,13,16,24,26,32}
çã
{8,13,16,24,26,32}
No.911ããããã¯ã¡ã¹ã4æ15æ¥ 16:50
ã¡ãã£ãšé¢çœãããšãæãã€ããŸããã
以äžã®ãããªæäœãããŠã¿ãŠãã ããã
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ãæ±ºããŠãã ããã
æèšç®ã§ãããªã p 㯠3 ã 5 ã 7 ãa 㯠2 以äžã§ a*p ã 100 ãè¶
ããªããããããããšæããŸãã
ã³ã³ãã¥ãŒã¿ã§ããå Žåã¯å¥œããªå€§ããã®æ°ã§ãèªç±ã«ã©ããã
1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ã€ãŸããp ã§å²ããš 1 äœãæ°ã a*p 以äžã®èªç¶æ°ã§ 1 ã€éžãã§ãã ããããšããããšã§ãã
p > 2 ã§ããã°ã
2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ãã£ãã®ãã€ã®äœã 2 ããŒãžã§ã³ã§ãã
p > 3 ã§ããã°ã
3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
äœã 3 ããŒãžã§ã³ã§ãã
以äžäœãã 1 ãã€å¢ãããªããç¹°ãè¿ããŠãäœã (p-1) ããŒãžã§ã³ãŸã§å®è¡ããŠãã ããã
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
(3) æåã®çµãæ°ãã«äœãçŽããåã³æäœ R ã p åç¹°ãè¿ããŠã¿ãŠãã ããã
ãããã¯ã3 ã€ãã4 ã€ãã®çµãäœã£ãŠããããã§ãã
ã»ãšãã©å
šãŠã®çµã§åãçŸè±¡ãèµ·ããããšã確èªããŠãã ããã
(4) ãå
šãŠã®çµãã§ã¯ãªããã»ãšãã©å
šãŠã®çµããšèšã£ãã®ã¯ãå®ã¯ a ãš p ãäºãã«çŽ ã ãš 1 ã€ã ãäŸå€ãããããã§ããããŠãããã¯ã©ããªçµã§ãäœãèµ·ããã§ãããïŒ
(5) ãããŸã§ã®å®éšã§ãæåã®æ¡ä»¶ãæºãããã㪠p-1 åçµã®ç·æ°ããp ã®åæ° +1 åããããšãçŽåŸããŠãããããšæããŸãã
ãšããã§ãæåã®æ°ã®éžã³æ¹ãããæãåºããŠããã®ãã㪠p-1 åçµã£ãŠäœéããããã§ããã£ãïŒ
No.859DD++4æ10æ¥ 21:54
DD++æ§ããã¯ããããããŸãã
ãã¥ãŒãã³ã®ããªã³ããã¢ã¯ãæç« ã°ããã§ãæ°åŒã¯ãªãã£ãããã§ãããããçŸä»£ç§éãç¿ãç©çåŠãæ°åŒäžå¿ã§ãããããã¯ãªã€ã©ãŒã®æ¥çžŸã®1ã€ã ããã§ãã
ããŠãDD++æ§ãæç« ã°ããã§ãªããæ°åŒãã¡ãã°ããŠãæžããŠãã ãããšãã£ãšãããããããªãããããªãããªïŒãšæããŸãã
ãªããšããªããªããã®ã§ããããïŒ
No.861ããããã¯ã¡ã¹ã4æ11æ¥ 08:32
ãã®è©±é¡ãæ°åŒã¯åŒãç®ãšæãç®ãšå°äœããåºãŠããŸããã
æãç® a*p ã¯æžããŠãŸãã
å°äœ k+np ãå
šéšæžããŠãŸãã
åŒãç®ã¯ãæç« äžã« 1 åããç»å ŽããŸããããããããã¡ãã¡æžããã»ããããã§ããïŒ
ããã以å€ãæžããããŠãèšç®ãååšããŸããã
No.862DD++4æ11æ¥ 08:40
ãããªããŸãããã©ãã§ãééããã®ã§ãããïŒ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ãæ±ºããŠãã ããã
æèšç®ã§ãããªã p 㯠3 ã 5 ã 7 ãa 㯠2 以äžã§ a*p ã 100 ãè¶
ããªããããããããšæããŸãã
ã³ã³ãã¥ãŒã¿ã§ããå Žåã¯å¥œããªå€§ããã®æ°ã§ãèªç±ã«ã©ããã
1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ã€ãŸããp ã§å²ããš 1 äœãæ°ã a*p 以äžã®èªç¶æ°ã§ 1 ã€éžãã§ãã ããããšããããšã§ãã
1+r1p (mod p)â¡1
p > 2 ã§ããã°ã
2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
ãã£ãã®ãã€ã®äœã 2 ããŒãžã§ã³ã§ãã
2+r2p (mod p)â¡2
p > 3 ã§ããã°ã
3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
äœã 3 ããŒãžã§ã³ã§ãã
3+r3p (mod p)â¡3
以äžäœãã 1 ãã€å¢ãããªããç¹°ãè¿ããŠãäœã (p-1) ããŒãžã§ã³ãŸã§å®è¡ããŠãã ããã
(p-1)+r(p-1)p (mod p)â¡p-1
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{r1,r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1)}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{r1,r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1),a*p}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{r2,r3,r4,r5,r6,ã»ã»ã»,r(p-1),a*p}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{r2-r1,r3-r1,r4-r1,r5-r1,r6-r1,ã»ã»ã»,r(p-1)-r1,a*p-r1}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
----ãããã¡ãããªãã
No.867ããããã¯ã¡ã¹ã4æ11æ¥ 15:01
ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
ã®ãšããã§ããã
éžãã æ°ã¯ rk ãããªããŠãk+rk*p ã®æ¹ã§ãã
å
·äœçãªæ°ã§ãããªããšãããã®ãå°ããé ãã䞊ã¹ãã®ã¯é£ãããšæããŸããã
No.868DD++4æ11æ¥ 15:11
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{1+r1p,2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1)}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{1+r1p,2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1),a*p}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{2+r2p,3+r3p,4+r4p,5+r5p,6+r6p,ã»ã»ã»,(p-1)+r(p-1),a*p}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{2+r2p-(1+r1p),3+r3p-(1+r1p),4+r4P-(1+r1p),5+r5p-(1+r1p),6+r6p-(1+r1p),ã»ã»ã»,(p-1)+r(p-1)p-(1+r1p),a*p-(1+r1p)}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
---ã¯ããããã§ãæç€ºã®æé ã¯ããã£ãŠãŸããã
ãããããå®éã®æ°ã§ãåé¡ãé²ããããã§ããïŒ
No.870ããããã¯ã¡ã¹ã4æ11æ¥ 15:38
ãããéžãã§äžŠã¹ããšããããå
·äœçãªæ°ã§ã©ããã
ã¯ã¡ã¹ããã㯠1+r1*p ãæå°å€ãšããŠããŸãããå®é㯠r1 ãš r2 ã®å€§å°ã«ãã£ãŠã¯
2+r2*p < 1+r1*p
ãšãªãå Žåããããå¿
ããã 1+r1*p ãå
é ã«ãããããããªãã®ã§ã
No.871DD++4æ11æ¥ 15:58
ã¯ã¡ã¹ãããããã£ãããã«
>ãã¥ãŒãã³ã®ããªã³ããã¢ã¯ãæç« ã°ããã§ãæ°åŒã¯ãªãã£ãããã§ãããããçŸä»£ç§éãç¿ãç©çåŠãæ°åŒäžå¿ã§ãããããã¯ãªã€ã©ãŒã®æ¥çžŸã®1ã€ã ããã§ãã
ãŸããã®ããããå€ãã®é«åãªåŠè
ãããé¢ãã£ãŠããŸãã®ã§âŠâŠ
埡åèãŸã§ã«ã
æè³ æ¢è¿ª, ç§åŠå²å
¥é 18äžçŽãšãŒãããã®ååŠç ç©¶ : åŠè
ãã¡ã®äº€æµãšè«äº, ç§åŠå²ç ç©¶, 2014-2015, 53 å·», 272 å·, p. 473-, å
¬éæ¥ 2020/12/14, Online ISSN 2435-0524, Print ISSN 2188-7535, https://doi.org/10.34336/jhsj.53.272_473
, https://www.jstage.jst.go.jp/article/jhsj/53/272/53_473/_article/-char/ja
No.874Dengan kesaktian Indukmu4æ11æ¥ 17:19
éäžãŸã§äœæ¥ããŠãããããã¯ã¡ã¹ãããããã®åŸã©ããªã£ãã®ãããããŸããããæ°æ¥çµã¡ãŸããã®ã§çµå±ãããäœã ã£ãã®ããšãããã¿ãã©ã·ãã
ãŸããã¿ã€ãã«ã§æåããã»ãšãã©æžããŠãããããªãã®ã§ããã
å®ã¯ããããã§ã«ããŒã®å°å®çãããåæ°ãã䜿ãããšã§åŒå€åœ¢ãªãã«çŽæ¥èšŒæã§ããªãããšè©Šã¿ããã®ã§ãã
æåã«æç€ºããæ°ã®éžã³æ¹ã¯å
šéšã§ a^(p-1) éããããŸãã
ãã®ãã¡ãå
šãŠã®éžæã§ a ã®åæ°ãéžãã { a, 2a, 3a, âŠâŠ, (p-1)a } ãšããçµã¯å¯äžæäœ R ã§èªåèªèº«ã«ãªããŸãã
ïŒa ãš p ãäºãã«çŽ ã®ãšãããã®éžã³æ¹ãå¿
ãå¯èœïŒ
ãããŠæ®ãã® a^(p-1) - 1 åã®çµã¯ãåãæ¡ä»¶ãæºããå¥ã®çµãé ã«å·¡ã£ãŠã2 以äžã® p ã®çŽæ°ãååŸã«èªåèªèº«ã«åž°ã£ãŠããŸãã
ããããp ã¯çŽ æ°ãªã®ã§ãã2 以äžã® p ã®çŽæ°ã㯠p 以å€ã«ãããŸããã
ã€ãŸãããã®æäœ R ã§ãããããšç¹ããé¢ä¿ p å 1 ã°ã«ãŒãã« a^(p-1) - 1 åã®ãã®ããããªãããããªãåããããŸãã
ãã£ãŠãa ã p ãšäºãã«çŽ ã§ããã°ãa^(p-1) - 1 㯠p ã®åæ°ã§ããããšã瀺ããâŠâŠãããšããã»ã©ãã¡ããšæžããŠã¯ããŸãããããªãã»ã©ç¢ºãã«æãç«ã¡ããã ãšèšãããããã®ãªã¢ãã£ãã§ããŸããã
âŠâŠãšããããšãªã®ã§ããã
ã¿ãªããããäœã a^(p-1) åã®ãã®ãçšæããŠããããã 1 ã€ãåãé€ããšãæ®ããæŒããªã p åãã€ã®ã°ã«ãŒãã«ãããããããããªãã®ãæãã€ãããæ¯éæããŠãã ããã
å®éã«ãã£ãŠã¿ããšãç°¡åã«äœãããã«èŠããŠãã¡ããã¡ãé£ããã§ãã
No.890DD++4æ14æ¥ 16:51
DD++æ§ãããã°ãã¯ã
æšæ¥ã1åç®ãŸã§ããããŸãããïŒïŒïŒã®åé¡ãŸã§ããªãå
ãããããã§ãã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸããçŽ æ° p ããã³ãããšäºãã«çŽ ãªèªç¶æ° a ãæ±ºããŠãã ããã
p=7ãa=5ããšããŸãã
ã1, 1+p, 1+2p, âŠâŠ, 1+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
1+3p=22
ã2, 2+p, 2+2p, âŠâŠ, 2+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
2+2p=16
ã3, 3+p, 3+2p, âŠâŠ, 3+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
3+2p=17
ã4, 4+p, 4+2p, âŠâŠ, 4+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
4+3p=25
ã5, 5+p, 5+2p, âŠâŠ, 5+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
5+2p=19
ã6, 6+p, 6+2p, âŠâŠ, 6+(a-1)p ã®äžããèªç±ã« 1 ã€éžãã§ãã ããã
6+2p=20
ããã«ãã£ãŠãa*p 以äžã®èªç¶æ°ã p-1 åéžã³åºããŸãããã
ã§ã¯ãããããå°ããé ã«äžŠã¹ãŠäžã€ã®çµãšããŠãã ããã
{16,17,19,20,22,25}
å°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
{16,17,19,20,22,25,35}
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
{17,19,20,22,25,35}
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
{1,3,4,6,9,19}
ããŠãã§ã¯ã
(1) æ°ããã§ãã p-1 åã®æ°ã®çµã§ãããå®ã¯æåã®çµã®äœãæ¹ã®æ¡ä»¶ã«åœãŠã¯ãŸã£ãŠããŸããïŒ
ããªãã¡ã
ã»a*p 以äžã®èªç¶æ° p-1 åãå°ããé ã«äžŠãã§ãã
ã»p ã§å²ã£ãäœã㯠1 ãã p-1 ãŸã§ 1 åãã€
ã«ãªã£ãŠããŸããïŒ
{1,3,4,6,9,19}â¡{1,3,4,6,2,5}(mod p=7)
(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
(1)
{1,3,4,6,9,19,35}
{3,4,6,9,19,35}
{2,3,5,8,18,34}
{2,3,5,8,18,34}â¡{2,3,5,1,4,6}(mod p=7)
(2)
{2,3,5,1,4,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(3)
{1,2,3,4,5,6,35}
{2,3,4,5,6,35}
{1,2,3,4,5,34}â¡{1,2,3,4,5,6}(mod p=7)
(4)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(5)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(6)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
(7)
{1,2,3,4,5,6}
{1,2,3,4,5,6}
{2,3,4,5,6,35}
{1,2,3,4,5,34}}â¡{1,2,3,4,5,6}(mod p=7)
çã
{1,2,3,4,5,6}(mod p=7)
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãŸã¡ãã£ãŠãŸãïŒ
No.892ããããã¯ã¡ã¹ã4æ14æ¥ 18:45
ã1 åç®ã®çµæã㯠{2,3,5,8,18,34} ã§ããã
No.893DD++4æ14æ¥ 19:04
ãã(2) ã® 1 åç®ãã€ãŸãå
šäœã® 2 åç®ã®çµæã®ããšã§ãã
No.894DD++4æ14æ¥ 19:05
ïŒ(2) ãšããããšã¯ãæ°ããã§ããçµã«å¯ŸããŠæäœ R ãããäžåºŠè¡ãããšãã§ããŸãã
ãããŠã§ããçµã«ãŸãããäžåºŠãããã«ããäžåºŠã
æäœ R ã p åç¹°ãè¿ãããšããäœããèµ·ãããšæããŸãããããã¯äœã§ãããã
ããããã£ãã€ããã§ãããã»ã»ã»ã»
ééã£ãŠãŸãããïŒ
ïŒ(3) æåã®çµãæ°ãã«äœãçŽããåã³æäœ R ã p åç¹°ãè¿ããŠã¿ãŠãã ããã
ãããã¯ã3 ã€ãã4 ã€ãã®çµãäœã£ãŠããããã§ãã
ã»ãšãã©å
šãŠã®çµã§åãçŸè±¡ãèµ·ããããšã確èªããŠãã ããã
ã§ãããã¯ã©ããªããã ãããšæ¢ãŸã£ãã®ã§ãã
No.895ããããã¯ã¡ã¹ã4æ14æ¥ 20:01
{1,3,4,6,9,19,35} <- æåŸã« 35 ãã€ããïŒæäœ R ã® 1 ã€ãïŒ
{3,4,6,9,19,35} <- 1 ãåãèœãšããŠïŒæäœ R ã® 2 ã€ãïŒ
{2,3,5,8,18,34} <- å
šéšãã 1 ãåŒãããå®æïŒæäœ R ã® 3 ã€ãïŒ
{2,3,5,8,18,34}â¡{2,3,5,1,4,6}(mod p=7) <- äœãããã©ãã©ã確èªããã ã
ããŠãæäœ R ã®å®æåã¯ã©ãã§ãããïŒ
æ¬åœã« {2,3,5,1,4,6} ã§ããïŒ
No.896DD++4æ14æ¥ 20:50
ãã 1 åæäœ R ã®å®çŸ©ããã¡ããšèªãã§ãã ããã
No.899DD++4æ15æ¥ 07:11
ïŒå°ããé ã«äžŠãã§ãã p-1 åã®æ°ã®çµã«ã以äžã®ãã㪠3 ã€ã®æé ãããªãæäœ R ãããŸãã
ãŸããæ«å°Ÿã« a*p ãä»ãå ããäžæçã« p åçµãšããŸãã
å
é ã®æ°ããã£ããèŠããäžã§åãèœãšããŠãp-1 åçµã«æ»ããŸãã
p-1 åã®æ°ãããããããã£ãåãèœãšããæ°ãåŒãç®ããŸãã
ã§ãããã
ãšãããšã
ïŒ{2,3,5,8,18,34} <- å
šéšãã 1 ãåŒãããå®æïŒæäœ R ã® 3 ã€ãïŒ
ã§ããããïŒ
No.900ããããã¯ã¡ã¹ã4æ15æ¥ 08:55
ã¯ãããããæäœ R ã®çµæã§ãã
ã§ããããæ¬¡ã¯ {2,3,5,8,18,34} ããåºçºã«ãªããŸãã
No.901DD++4æ15æ¥ 08:58
(1)
{1,3,4,6,9,19,35}
{3,4,6,9,19,35}
{2,3,5,8,18,34}
(2)
{2,3,5,8,18,34}
{3,5,8,18,34,35}
{1,3,6,16,32,33}
(3)
{1,3,6,16,32,33}
{3,6,16,32,33,35}
{2,5,15,31,32,34}
(4)
{2,5,15,31,32,34}
{5,15,31,32,34,35}
{3,13,29,30,32,33}
(5)
{3,13,29,30,32,33}
{13,29,30,32,33,35}
{10,26,27,29,30,32}
(6)
{10,26,27,29,30,32}
{26,27,29,30,32,35}
{16,17,19,20,22,25}
(7)
{16,17,19,20,22,25}
{17,19,20,22,25,35}
{1,3,4,6,9,19}
çã
{1,3,4,6,9,19}
ããã§ããã£ãŠãŸãããïŒ
No.902ããããã¯ã¡ã¹ã4æ15æ¥ 09:14
ã¯ããååã§ãã£ãŠããããšã¯ãã£ãŠããŸãã
ãã ã{16,17,19,20,22,25} ãã {1,3,4,6,9,19} ãäœã£ãã®ã 1 åç®ã§ãããã
> (6)
> {10,26,27,29,30,32}
> {26,27,29,30,32,35}
> {16,17,19,20,22,25}
ã 7 åç®ã§ãããã§ã¹ãããã§ãã
No.903DD++4æ15æ¥ 09:20
é¢çœãã£ãã®ã§ã玹ä»ããããŸãã
Hereâs How Two New Orleans Teenagers Found a New Proof of the Pythagorean Theorem | by Keith McNulty | Apr, 2023 | Medium
h_TT_ps://keith-mcnulty.medium.com/heres-how-two-new-orleans-teenagers-found-a-new-proof-of-the-pythagorean-theorem-b4f6e7e9ea2d
No.839Dengan kesaktian Indukmu4æ9æ¥ 13:04
Dengan kesaktian Indukmuæ§ãããã°ãã¯ã
google翻蚳ããŸããã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
éšåã®ãã 1 ã€ã®çç±ã¯ããããã®è¥ãå
é§è
ãææ¡ãã蚌æãã確ç«ãããæ°äººã®æ°åŠè
ã«åœŒãã®èšèãé£ãç©ã«ãããããããªããšããããšã§ãã
ããã¯ã圌ãã®èšŒæãäžè§æ³ã䜿çšããŠããããã§ãã
ã§ã¯ããªãããããããªã«å€§ããªåé¡ãªã®ã§ããïŒ ããŠãç§ãã¡ã®äžè§æçåŒãšæ³åã®å€ãã¯ãã¿ãŽã©ã¹ã®å®çã«äŸåããŠãããããå€ãã®æ°åŠè
ã¯ãäžè§æ³ã䜿çšããå®çã®èšŒæã¯åŸªç°è«çã§ãããšç€ºåããŠããŸãã å¥ã®èšãæ¹ãããã°ãäžè§æ³ã䜿çšããŠãã¿ãŽã©ã¹ã蚌æããããšã¯ãåºæ¬çã« A ã䜿çšã㊠B ã蚌æããããšã§ãããA ãæ¢ã« B ã«äŸåããŠããå Žåã«ã圌ãã¯äž»åŒµããŸãã -å®çã®äžè§æ³ã«ãã蚌æãããã³äžè§æ³ã®èšŒæã¯äžå¯èœã§ããããšãæç€ºçã«è¿°ã¹ãŠããŸãã
ãããããã®èгç¹ã¯ããæ°å幎ã§ãŸããŸãçåèŠãããŠããŠããããã以æ¥ããã¿ãŽã©ã¹ã®ããã€ãã®äžè§æ³ã«ãã蚌æãè¡ãããŠããŸãã. ãžã§ã³ãœã³ãšãžã£ã¯ãœã³ã®èšŒæããã¿ãŽã©ã¹ã®æåã®äžè§æ³ã«ãã蚌æã§ãããšããã¡ãã£ã¢ã®äž»åŒµã¯èªåŒµãããŠããŸããã圌ãã®èšŒæã¯ããããŸã§ã«èŠãäžã§æãçŸãããæãåçŽãªäžè§æ³ã®èšŒæã§ããå¯èœæ§ãååã«ãããæããã«è¥ããŠéãç¥æ§ã®äœåã§ããã å€ãã®çµéšè±å¯ãªæ°åŠè
ã®ä»äºãç¹åŸŽä»ããæ·±ãç ç©¶ã®å¹Žã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
æ°ããææ³ãªãã ãã©ãèŠããã«ã
å¥ã®èšãæ¹ãããã°ãäžè§æ³ã䜿çšããŠãã¿ãŽã©ã¹ã蚌æããããšã¯ãåºæ¬çã« A ã䜿çšã㊠B ã蚌æããããšã§ãããA ãæ¢ã« B ã«äŸåããŠããå Žåã«ã圌ãã¯äž»åŒµããŸãã -å®çã®äžè§æ³ã«ãã蚌æãããã³äžè§æ³ã®èšŒæã¯äžå¯èœã§ããããšãæç€ºçã«è¿°ã¹ãŠããŸãã
ãšããããšããåé¡ç¹ã§ããããšããããšã§ããã
No.847ããããã¯ã¡ã¹ã4æ9æ¥ 19:37
ã¯ãã
ã§ãã®ã§ä»åã¯ãããã埪ç°è«æ³ãé¿ããŠããããšããçè§£ã§ããããããšããšã
No.848Dengan kesaktian Indukmu4æ10æ¥ 07:26
ãããããŒãŒãŒãèªããšåŸªç°è«æ³ã®å®å
šãªåé¿ã¯åŸ®åŠã«å€±æããŠãŸãã
In this case we have an isoceles right-angled triangle and, our angles ⺠= β = Ï/4 radians. So our hypotenuse is a/sin(Ï/4) = â2a, which satisfies the Pythagorean Theorem.
ãã® sin(Ï/4) ã®å€ã¯ã©ãããïŒ
éåžžã¯äžå¹³æ¹ã®å®çã§å°åºãããã®ã§ãããããã埪ç°è«æ³ãé¿ããããšã«æåãããšäž»åŒµããã«ã¯ãããå¥ã®æ¹æ³ã§å°åºããŠèŠããå¿
èŠããã£ãã§ãããã
ãŸããçžäŒŒãªå³åœ¢ã®é¢ç©æ¯ã䜿ããšãåé¿æ¹æ³ã¯ãããã§ãããã®ã§å çä¿®æ£ã¯å®¹æã§ããããã©ã
No.849DD++4æ10æ¥ 11:50
ããã£ã
a=b ã®ç¹æ®ãªã±ãŒã¹ã§ã¯
AãCã®åæ¯ã 0 ã«ãªã£ãŠããŸã
ä»åã®èšŒæãéçšããªãã
ããããè©±ã®æµããªã®ã§ããïŒïŒ
ãã®ã±ãŒã¹ã§ã®ç蚌æã«
sin(Ï/4)
ã䜿ãã®ã¯ç¢ºãã«ååã§ããã
No.850Dengan kesaktian Indukmu4æ10æ¥ 12:58
å¯ããããŠããã³ã¡ã³ãã远ãããããšãã
a = b ã®ã±ãŒã¹ã§ã¯ã次ã®ããã«æ¹åããæ¡ããããããŠããŸããã
By the way, that case is trivial: the triangle is a one-fourth of a square whose side length is $c$. The area of this square is c^2, while the triangleâs area is (ab)/2 = a^2/2. Therefore c^2=4 times a^2/2 = 2a^2 = a^2+b^2, as desired.
No.851Dengan kesaktian Indukmu4æ10æ¥ 13:46
google翻蚳ããŸãã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ãã®æ°ãã蚌æ ã¯äœã§ããïŒ
ããããŸããã®ã§ããããç§ãèããæ¹æ³ã§ãã ãã®å³ãèŠãŠããã®èšŒæãéããŠãã£ãšåç
§ããŠãã ããã
å³ã®ããã«ã蟺 aãbãc (æèŸº) ãæã€å·Šäžã®åçŽãªçŽè§äžè§åœ¢ããå§ããŸãããã ããã§ã¯ãa â b ãšä»®å®ããŸããã â ãã®èšäºã®æåŸã®æ³šã§ãa = b ã®èªæãªç¹æ®ãªã±ãŒã¹ãæ±ããŸãã é·ã b ãš c ã®èŸºã®éã®è§åºŠã ⺠ãšããé·ã a ãš c ã®èŸºã®éã®è§åºŠã β ãšããŸãã æ¬¡ã«ããã®å
ã®çŽè§äžè§åœ¢ãã 3 ã€ã®å¹ŸäœåŠçãªã¹ããããäœæããŸãã
é·ã b ã®èŸºã«åæ ããŠãå³äžã®ç䟡äžè§åœ¢ã圢æããŸãã
å
ã®äžè§åœ¢ã®é·ã c ã®èŸºã«åçŽãªçŽç·ãå»¶é·ããŸãã
åå°ããäžè§åœ¢ã®æèŸºããé£ç¶ç·ãå»¶é·ããŸãã
ã¹ããã 2 ãš 3 ããå»¶é·ããç·ã亀ãããšãå³ã®ããã«ãæèŸºã®é·ãã Cã蟺ã A ãš c ã®æ°ãã倧ããªçŽè§äžè§åœ¢ã圢æãããŸãã
ãã®å€§ããªçŽè§äžè§åœ¢å
ã«ãå³ã®ããã«äžé£ã®å°ãããŠå°ããé¡äŒŒã®çŽè§äžè§åœ¢ãæç»ãããµã€ãºãæžå°ããé¡äŒŒã®äžè§åœ¢ã®ç¡éã®ã·ãŒã±ã³ã¹ã圢æããŸãã
ãã®ç¡éã®çžäŒŒäžè§åœ¢ã®ã·ãŒã±ã³ã¹ã䜿çšããŠãé·ã A ãš C ãå°åºããæ¹æ³ã調ã¹ãŸãã
å°ããäžè§åœ¢ã®èŸºã®é·ããå°ãåºã
蟺ã®é·ã A ã®å·Šäžãã 1 çªç®ã®çŽè§äžè§åœ¢ãèŠããšããã®äžè§åœ¢ã®èŸºã®é·ã㯠2a ã§ããããããã£ãŠæèŸºã®é·ã㯠2a/sinβ ã§ãã ããããå
ã®äžè§åœ¢ãããsinβ = b/c ã§ããããšãããã£ãŠããã®ã§ããã®æèŸºã¯é·ã (2ac)/b ã§ãããšçµè«ä»ããããšãã§ããŸãã ããã«ããããã®äžè§åœ¢ã® 3 çªç®ã®èŸºã¯ 2a²/b ã«ãªããŸãã
ããã«å³åŽã®äžè§åœ¢ã«ç§»åãããšãç蟺㮠1 ã€ãé·ã 2a²/b ã§ããããšãããããŸãããããã£ãŠããã®äžè§åœ¢ã®æèŸº (蟺ã®é·ã C ã®ã»ã°ã¡ã³ã) 㯠2a²/(bsinβ) = (2a²c) /b² ã§ãã
ãã®ããã»ã¹ãç¶ããããšãã§ããŸãããå°ããªçžäŒŒäžè§åœ¢ã®ããããã a²/b² ã®ä¿æ°ã§æžå°ããããšãæããã«ãªããŸãã ããã¯ãé·ã A ãæåã®é
(2ac)/b ãšå
¬æ¯ a²/b² ãæã€çæ¯çŽæ°ã§ããããšãæå³ããŸãã åæ§ã«ãé·ã C 㯠c ã§å§ãŸããæåã®é
(2a²c)/b² ãšå
¬æ¯ a²/b² ã®çæ¯çŽæ°ã«ãªããŸãã
é·ã A ãš C ã®èšç®
ããã§ãçæ¯çŽæ°ã®åã®åŒã䜿çšããŠãé·ã A ãš C ãèšç®ã§ããŸããåé
k ãšå
¬æ¯ r ã®çæ¯çŽæ°ã®åã®åŒã¯ãk/(1-r) ã§ãã ãã®åèšã¯ãr ã®çµ¶å¯Ÿå€ã 1 æªæºã®å Žåã«åæããŸãããã®å Žåãr 㯠a²/b² ã§ãããããåžžã«åæããããšã確èªã§ããŸã (a>b ã®å Žåã¯ããããã亀æããã ãã§ã)ã
ããã§ã¯ãé·ã A ãèšç®ããŠã¿ãŸãããããã®å Žåãk = (2ac)/b ããã³ r = a²/b² ãšãªãã®ã§ã
A=2ac/b(1-a^2/b^2)=2abc/b^2-a^2
k = (2a²c)/b² ã§åæ§ã®ã¢ãããŒãã䜿çšããæåã« c ã远å ããå¿
èŠãããããšãæãåºããŠãã ããã
C=c+2a^2c/b^2(1-a^2/b^2)=c(b^2+a^2)/(b^2-a^2)
ããã§ã¯ãé·ã A ãèšç®ããŠã¿ãŸãããããã®å Žåãk = (2ac)/b ããã³ r = a²/b² ãšãªãã®ã§ã
k = (2a²c)/b² ã§åæ§ã®ã¢ãããŒãã䜿çšããæåã« c ã远å ããå¿
èŠãããããšãæãåºããŠãã ããã
ãã®çŸãã蚌æãç· ãããã
A ãš C ã®æ¯ãåããšã©ããªããèŠãŠã¿ãŸãããã
A/C=2ab/(a^2+b^2)
ããããå
ã®å³ããããã㯠sin(2âº) ã§ããããšãããããŸãã
ããã§ãå
ã®çŽè§äžè§åœ¢ãåæ ããŠåœ¢æãããäžã®äºç蟺äžè§åœ¢ã®æ£åŒŠèŠåãèŠãŠã¿ãŸãããã æ£åŒŠå®çã¯çŽè§äžè§åœ¢ã«äŸåããªãããšã«æ³šæããŠãã ããã ãµã€ã³ ã«ãŒã«ã¯ãã©ã®äžè§åœ¢ã§ãã蟺ãšãã®å察åŽã®è§åºŠã®ãµã€ã³ãšã®æ¯çã¯åžžã«åãã§ãããšè¿°ã¹ãŠããŸãã
ãããã£ãŠïŒ
sin 2α/2a=sinβ/c
ãããã£ãŠãçŸåšããã£ãŠããããšã次ã®åŒã«å€æããŸãã
b/(a^2+b^2)=b/c^2
ãã®ç¶æ³ã§ã¯ãaãbãc ã®ãããããŒãã§ã¯ãªãããšã«æ³šæããååãåäžã§ããããšã«æ³šæãããšã忝ãåäžã§ãããšããçµè«ã«è³ããŸãã ããã§ãã¿ãŽã©ã¹ã®å®çã蚌æãããŸããã
[泚: a = b ãšããç¹æ®ãªã±ãŒã¹ã§ã¯ãäžè§åœ¢ã«é·ã a ã® 2 ã€ã®çãèŸºãšæèŸºãããå Žåã蚌æã¯èªæã§ãã ãã®å ŽåãçŽè§äºç蟺äžè§åœ¢ããããè§åºŠ ⺠= β = Ï/4 ã©ãžã¢ã³ã§ãã ãããã£ãŠãæèŸºã¯ a/sin(Ï/4) = â2a ã§ããããã¿ãŽã©ã¹ã®å®çãæºãããŸãã ããã£ã¢ã ãŠãŒã¶ãŒã«æè¬
ãŠã©ããã·ãã
ãã®ç¹å¥ãªã±ãŒã¹ã«å¯ŸåŠããå¿
èŠæ§ãææããŠãããŠ.]
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
å³ãããã«ã¯ãããŸããããåæã«ã¯ãããŸãã
No.852ããããã¯ã¡ã¹ã4æ10æ¥ 14:20
æŸããã®ã§ãã
ððð ðŠ = ððð (ð¥ â (ð¥ â ðŠ))
= ððð ð¥ ððð (ð¥ â ðŠ) + ð ðð ð¥ ð ðð(ð¥ â ðŠ)
= ððð ð¥(ððð ð¥ ððð ðŠ + ð ðð ð¥ ð ðð ðŠ) + ð ðð ð¥(ð ðð ð¥ ððð ðŠ â ððð ð¥ ð ðð ðŠ)
= (ððð ² ð¥ + ð ðð² ð¥) ððð ðŠ.
cos y ã0ã§ãªããã°ã䞡蟺ãããã§å²ãã
以äžã¯ããã¿ãŽã©ã¹ã®å®çãéœã«ã¯äœ¿ããã«
cos^2x ãš sin^2x ãšã®åã 1 ãšç€ºããã®ã§ãã
No.853Dengan kesaktian Indukmu4æ10æ¥ 16:25
å æ³å®çã¯äžäœã©ãããæ¹§ããŠåºãã®ã§ããããã
sin ã cos ã埮åã§å®çŸ©ããŠããå Žåã¯å æ³å®çããã¯ããŒãªã³å±éãšäºé
å®çã§ç€ºãããšã«ãªãã§ããããããã®å Žåã¯ããã§å€§äžå€«ãšãã話ãªã®ããªïŒ
No.854DD++4æ10æ¥ 16:45
ãšæããŸããããå°ãèããŠã¿ãã
sâ(x) = c(x)
câ(x) = -s(x)
s(0) = 0
c(0) = 1
ã®è§£ã s(x) = sin x, c(x) = cos x ãšããå®çŸ©ã®å Žåã
{ (sin x)^2 + (cos x)^2 }â = 0 ãäžç¬ã§ç€ºããã®ã§ãå æ³å®çã䜿ããŸã§ããªãã£ãâŠâŠã
çŽæ°ã§å®çŸ©ããå Žåããã®åŸ®åã®é¢ä¿åŒãããäœããŸãããäœãç®çãšããåŒå€åœ¢ã ã£ããã§ããããã³ã¬ã
No.855DD++4æ10æ¥ 16:56
倱瀌ããŸããã
æŸã£ãå ŽæãèšãæŒãããŸããã
https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf
ã§ãã
No.856Dengan kesaktian Indukmu4æ10æ¥ 17:07
ããéè§éå®ãªãã§ããã
ãããªã確ãã«å æ³å®çã«äžå¹³æ¹ã¯äžèŠã§ããã
No.857DD++4æ10æ¥ 17:17
ã§ããéè§éå®ã§ããã£ã±ãå æ³å®ç䜿ããŸã§ããªããããªæ°ãããŸããã
â B = Ξ, â C = Ï/2, AB = 1 ã®äžè§æ¯ã®å®çŸ©ã«äœ¿ããã€ãã®çŽè§äžè§åœ¢ ABC ã«å¯Ÿãã
蟺 DE ãç¹ C ãéãããã«é·æ¹åœ¢ ABDE ãæžãã°çžäŒŒãªçŽè§äžè§åœ¢ã 3 ã€ã§ããŠã
BC = cosΞ 㧠DC = (cosΞ)^2
AC = sinΞ 㧠EC = (sinΞ)^2
(cosΞ)^2 + (sinΞ)^2 = DC + EC = AB = 1
ã§çµãã話ãªãããªã
No.858DD++4æ10æ¥ 17:34
Dengan kesaktian Indukmuæ§ããã¯ããããããŸãã
ïŒã§ãã®ã§ä»åã¯ãããã埪ç°è«æ³ãé¿ããŠããããšããçè§£ã§ããããããšããšã
ããããsinαãsinβãšããäžè§é¢æ°ã®æŠå¿µã䜿ã£ãŠããŸãããã
ïŒèŸºã®é·ã A ã®å·Šäžãã 1 çªç®ã®çŽè§äžè§åœ¢ãèŠããšããã®äžè§åœ¢ã®èŸºã®é·ã㯠2a ã§ããããããã£ãŠæèŸºã®é·ã㯠2a/sinβ ã§ãã ããããå
ã®äžè§åœ¢ãããsinβ = b/c ã§ããããšãããã£ãŠããã®ã§ããã®æèŸºã¯é·ã (2ac)/b ã§ãããšçµè«ä»ããããšãã§ããŸãã ããã«ããããã®äžè§åœ¢ã® 3 çªç®ã®èŸºã¯ 2a²/b ã«ãªããŸãã
ãããæããã§ãããsinβ = b/c ã¯ã
ïŒããã«å³åŽã®äžè§åœ¢ã«ç§»åãããšãç蟺㮠1 ã€ãé·ã 2a²/b ã§ããããšãããããŸãããããã£ãŠããã®äžè§åœ¢ã®æèŸº (蟺ã®é·ã C ã®ã»ã°ã¡ã³ã) 㯠2a²/(bsinβ) = (2a²c) /b² ã§ãã
ã§ã䜿ãããŠããŸããb/cãsinβãšããããšãªãã«ãè«çã¯ãã¿ããŠãããªããšæããŸãã
ïŒA ãš C ã®æ¯ãåããšã©ããªããèŠãŠã¿ãŸãããã
A/C=2ab/(a^2+b^2)
ããããå
ã®å³ããããã㯠sin(2âº) ã§ããããšãããããŸãã
ããã§ããsin(2âº)ãšããäžè§é¢æ°ã®æŠå¿µã䜿ãããŠããŸãã
ïŒåŸªç°è«æ³ãé¿ããŠãã
ãšã¯èšããªãã®ã§ã¯ãªãã§ããããïŒ
ã§ãªããã°ãæç« ã®ååã¯ããããªãã£ãã¯ãã§ããã埪ç°è«æ³ãªããã©ãããšããå眮ããããããããã®ææžãããã®ã§ã¯ãªãã§ããããïŒ
No.860ããããã¯ã¡ã¹ã4æ11æ¥ 07:23
äžè§é¢æ°ã®å®çŸ©èªäœã䜿ãã ããªã埪ç°è«æ³ã«ã¯ãªããŸãããã
ãæå
ã®æ°åŠã®æç§æžãèŠãŠãã ããã
(sinΞ)^2 + (cosΞ)^2 = 1 ãšããéèŠãªåŒã®èšŒæã«äžå¹³æ¹ã®å®çãé¢ãã£ãŠããŸããã
ã ããããã®ããŒãžããåŸã«æžããŠããããšã¯ãäžå¹³æ¹ã®å®çã®èšŒæã«äœ¿ã£ãŠã¯ãªããŸããã
èšãæ¹ãå€ããã°ããã®ããŒãžããåã«æžããŠããããšã¯å¥ã«äœ¿ã£ãŠãäœãåé¡ã¯ãªããã§ãã
ã ãããäžè§é¢æ°ïŒã®äžå¹³æ¹ã®å®çã䜿ãåã®éšåïŒã§ãã¡ããšäžå¹³æ¹ã®å®çã蚌æããããšã«æåããããšãã話é¡ãªã®ã§ããã
ã¯ã¡ã¹ãããã¯ã©ããæ°åŒããèŠãŠããªãããã§ãããèšèªãšåãããŠèªãããã«ããæ¹ãããããããšã
No.863DD++4æ11æ¥ 08:54
(sinΞ)^2 + (cosΞ)^2 = 1
ãããåæãšããã«
æ£åŒŠå®çã£ãŠãªããã€ãã ã£ãïŒ
ãããã念ã®ããã«ç¢ºèªã¯ããããŸããã
埪ç°è«æ³ã«ãªã£ãŠããªãããšã確èªããããã§ãã
No.864Dengan kesaktian Indukmu4æ11æ¥ 09:08
ããããããã°æ£åŒŠå®ç䜿ã£ãŠãŸãããã
ã§ã¯ã2 ã€äžã®ã³ã¡ã³ããèšæ£ã
ãäžè§é¢æ°ïŒã®äžå¹³æ¹ã®å®çã䜿ãåã®éšåãšãæç§æžã®æ²èŒé çã«ã¯åŸãã ãã©äžå¹³æ¹ã䜿ããã®ã«äŸåãã蚌æãæ§æãããŠãããã®ïŒã§ãã¡ããšäžå¹³æ¹ã®å®çã蚌æããããšã«æåããã
ã§ããã
éè§ã®äžè§æ¯ã®å®çŸ©
tan = sin / cos
éè§ã®æ£åŒŠå®ç
第äžäœåŒŠå®çïŒé«æ ¡ã§ç¿ããã€ã¯ç¬¬äºäœåŒŠå®çã§ããã£ã¡ã¯ãã¡ïŒ
éè§ã®å æ³å®çâŠâŠãããã§ããã
No.865DD++4æ11æ¥ 09:25
ãªããæããã§äŒŒããããªããšãã£ããããªããšæã£ãŠèª¿ã¹ãŠã¿ãããããŸããã
ãäžç·å®çã
http://shochandas.xsrv.jp/mathbun/mathbun658.html
äžå¹³æ¹ã®å®çã䜿ããã«äžç·å®çã瀺ãããããšãã話é¡ã§ãã
ååã¯ãªãã¯ãããé£ã¹ãããã«ããŠãããŸãã
No.866DD++4æ11æ¥ 09:46 管ç人ããããç矩ãããã£ãŠããããšã«
æ°ãä»ããŸãããæŠããäžèšã®ããšããšåããŸãã
ãæ£åŒŠå®çã¯(第ïŒ)äœåŒŠå®çããå°ãããããã®äœåŒŠå®çããã¯çŽæ¥ã«äžå¹³æ¹ã®å®çãå°ãããããžã§ã³ãœã³ãšãžã£ã¯ãœã³ã«ããä»åã®æ°èšŒæãæ£åŒŠå®çã«äŸæ ããã®ã¯ãŸããã®ã§ã¯ãªããïŒã
ç§ãå°ã
èã蟌ã¿ãŸããã
調ã¹ããšããã以äžãããããŸãããããªãã¡
äžè§é¢æ°ã®å æ³å®çã®ããšã§ã¯
æ£åŒŠå®çãšç¬¬ïŒäœåŒŠå®çãšç¬¬ïŒäœåŒŠå®çãšã¯
åå€ã§ããã
ã§ããã
å æ³å®çã«äŸæ ãã€ã€
ãžã§ã³ãœã³ãšãžã£ã¯ãœã³ãšã®æ°èšŒæãçè§£ããããšãããã¯ãŸããããã埪ç°è«æ³ã®è¬ããå
ããªãã
ãããã©ãããã°ããã®ãïŒ
æ£åŒŠå®çã®åçš®ã®èšŒæã調ã¹ãŠãã ãã£ãŠããããã¹ããã¿ã€ããŸããã
http://izumi-math.jp/K_Satou/seigen/seigen.htm
äžèšã®ãªãããé©åãªãã®ãèŠãã ããã°ãããšæããŸããã
ãžã§ã³ãœã³ãšãžã£ã¯ãœã³ã¯
é¢ç©ã«ãã蚌æãåé¿ã
蟺ã®é·ãã®çžäŒŒãå©çšããŠããŸããã
ãã®å¿åãšãæŽåããã»ããè¯ãã§ããã
ãããã®ææžã®ãæ®éã®èšŒæïŒïŒïŒã
ãè§£ããããããšæããŸããã
ãããã¯ã
ãïŒã幟äœåŠç蚌æãïŸïœ¯ïŸïœ°ïŸïŸã«ãããäžè§æ³ã®åºç€ãã€ãã£ãïŸïœ·ïŸïœµïŸïŸïŸïŸïœœ(1436-1376)ã®èšŒæã
ãçŽ æµã§ãã
ããã«ããŠããDD++ããã«ãã
No.858
ã¯ã¯ãŒã«ã§ããã
No.885Dengan kesaktian Indukmu4æ13æ¥ 11:13
æå
ã«é«æ ¡æ°åŠã®æç§æžããªãã®ã§èšæ¶é Œãã§ãããæ¥æ¬ã®é«æ ¡æè²ã§ã¯æ£åŒŠå®çã¯åãååšè§ããã€çŽè§äžè§åœ¢ã䜿ã£ãŠèšŒæããŠãããšæããŸãã
ããã§äœ¿ã£ãŠããã®ã¯
ã»ååšè§ã®å®ç
ã»ã¿ã¬ã¹ã®å®ç
ã»äžè§æ¯ã®å®çŸ©
ã»çŽåŸãšååŸã®å®çŸ©
ãããã ã£ãã¯ãã§ãäžå¹³æ¹ã®å®çã¯å
šã䜿ã£ãŠãªãã§ããã
äœåŒŠå®çãäœ¿ãæ¹æ³ã¯ãç§ã¯ä»å調ã¹ãŠåããŠååšãç¥ããŸããã
æµ·å€ã ãšã©ã®æ¹æ³ã§ã®èšŒæãã¡ãžã£ãŒãªãã ããã
No.886DD++4æ13æ¥ 12:08
law of sines ã§æ€çŽ¢ããŠè±èªããŒãžãããããèŠãŠã¿ãŸããã
æ¥æ¬ãšã¯éã£ãŠã=2R ãã€ããŠããªã圢ã§ç޹ä»ãããŠããããšãã»ãšãã©ã®ããã§ãã
ãã®ããã蚌æã以äžã§çµãããšããã®ãæ®éã®ããã
A ã B ãéè§ã®å Žåãé ç¹ C ãã蟺 AB ã«äžãããåç·ã®é·ããèãããš
b*sinA = a*sinB
䞡蟺ã sinA ããã³ sinB ã§å²ã£ãŠåŸãããã
ã©ã¡ãããéè§ã®å Žåã¯å
è§ãšå€è§ã® sin ã®å€ã¯çããããšãèããã°åãããšãèšããã
åé¡ã¯äžå¹³æ¹ã®å®çã䜿ã£ãŠãããã©ããã
çŽè§ãéè§ã®å Žåã¯ããããäžå¹³æ¹ã®å®çã䜿ã£ãŠå®çŸ©ããã®ã§ãã¡ã§ãããéè§ã®å Žåã«éå®ããã°äœ¿ã£ãŠãªãã§ããã
ãããŠä»¶ã®èšŒæã§ã¯ã2αãβãéè§ã§ãã
ãã£ãŠãæµ·å€ã§äž»æµã£ãœãæ£åŒŠå®çã®èšŒææ¹æ³ãåæãšããå Žåã埪ç°è«æ³ã«ã¯ãªã£ãŠããªããšèšã£ãŠããããã§ãã
No.889DD++4æ13æ¥ 21:11
äºé
å®çã¯ã
(a+b)^n=nC0 a^n b^0+nC1 a^(n-1) b^1+nC2 a^(n-2) b^2+nC3 a^(n-3) b^3+ã»ã»ã»ã»+nC(n-1) a^(n-(n-1)) b^(n-1)+nCn a^(n-n) b^n---(1)
ã§ã
ããããn
(a+b)^n=Σ{nCi a^(n-i) b^i}----(2)
ããããi=0
ãšãæžããŸãã
ã§ã¯ã
(1+1)^n=nC0 1^n 1^0+nC1 1^(n-1) 1^1+nC2 1^(n-2) 1^2+nC3 1^(n-3) 1^3+ã»ã»ã»ã»+nC(n-1) 1^(n-(n-1)) 1^(n-1)+nCn 1^(n-n) 1^n
2^n=nC0+nC1+nC2+nC3+ã»ã»ã»+nC(n-1)+nCn----(3)
ãšããæåãªå
¬åŒã«ããã©ãçããŸãã
ãŸãã
(a+b)^n=nC0 a^n b^0+nC1 a^(n-1) b^1+nC2 a^(n-2) b^2+nC3 a^(n-3) b^3+ã»ã»ã»ã»+nC(n-1) a^(n-(n-1)) b^(n-1)+nCn a^(n-n) b^n
(a+b)^n=a^n +nC1 a^(n-1) b^1+nC2 a^(n-2) b^2+nC3 a^(n-3) b^3+ã»ã»ã»ã»+nC(n-1) a b^(n-1)+b^n----(4)
(a+b)^n=a{a^(n-1)n +nC1 a^(n-2) b^1+nC2 a^(n-3) b^2+nC3 a^(n-4) b^3+ã»ã»ã»ã»+nCn-1) b^(n-1)}+b^n
(a+b)^n=aA+b^n ----(5)
ãã ããA=a^(n-1)n +nC1 a^(n-2) b^1+nC2 a^(n-3) b^2+nC3 a^(n-4) b^3+ã»ã»ã»ã»+nCn-1) b^(n-1)
(a+b)^nã¯ã(5)åŒãšãæžããŸãã
ïŒïŒäºé²å°æ°ã®10é²å°æ°åã®åé¡
ããã§ãäºé²å°æ°ã®10é²å°æ°åãèããŠã¿ãŸãããã1/2=0.5ã1/2^2=1/4=0.25ã1/8=0.125ã1/16=0.0625ã»ã»ã»1/256=0.00390625
ãšæ«å°Ÿããå¿
ã5ã«ãªãã®ã§ãã蚌æããŠã¿ãŸãããã
1/(2^n)=1/(2^n) 10^n/10^n=1/(2^n) (2^nã»5^n)/10^n=5^n/10^n=5(4+1)^(n-1)/10^n
ããã§ãäºé
å®çã®(5)åŒããã(4+1)^(n-1)=4A+1
5(4+1)^(n-1)/10^n=5(4A+1)/10^n=(20A+5)/10^n=20A/10^n+5/10^n=2A/10^(n-1)+5/10^n
ããã§ã2Aã¯èªç¶æ°ãªã®ã§ã5/10^nãããäžäœã®å°æ°ã§ãã
ãããã£ãŠãæ«å°Ÿã¯5ã«ãªããŸãã[蚌æçµãã]
äžè¬ã«2é²å°æ°ã¯
n
Σ{ai(1/2^i)} ãã ããaiã¯0ã1
i=0
ãªã®ã§ããã¹ãŠã®2é²å°æ°ã®10é²å°æ°åã¯æ«å°Ÿã¯5ã«ãªããŸãã
ãããããäºé²å°æ°ã§ã¯10é²å°æ°ã®1/5=0.2ã¯è¡šããªãããšã«ãªããŸããã0.24ã0.23ã0.22ã0.21ã衚ããŸããã
No.824ããããã¯ã¡ã¹ã4æ8æ¥ 14:26
ïŒïŒÎ±^n=nB+αã®å°åº
(4)åŒããã
ãã(1+1)^n=1^n +nC1 1^(n-1) +nC2 1^(n-2)+nC3 1^(n-3)+ã»ã»ã»ã»+nC(n-1) 1+1
ãã(2+1)^n=2^n +nC1 2^(n-1) +nC2 2^(n-2)+nC3 2^(n-3)+ã»ã»ã»ã»+nC(n-1) 2+1
ãã(3+1)^n=3^n +nC1 3^(n-1) +nC2 3^(n-2)+nC3 3^(n-3)+ã»ã»ã»ã»+nC(n-1) 3+1
ãã(4+1)^n=4^n +nC1 4^(n-1) +nC2 4^(n-2)+nC3 4^(n-3)+ã»ã»ã»ã»+nC(n-1) 4+1
ãã(5+1)^n=5^n +nC1 5^(n-1) +nC2 5^(n-2)+nC3 5^(n-3)+ã»ã»ã»ã»+nC(n-1) 5+1
ããããããããããããããããããããã»
ãã(r+1)^n=r^n +nC1 r^(n-1) +nC2 r^(n-2)+nC3 r^(n-3)+ã»ã»ã»ã»+nC(n-1) r+1
ããããããããããããããããããããã»
+)ã(a+1)^n=a^n +nC1 a^(n-1) +nC2 a^(n-2)+nC3 a^(n-3)+ã»ã»ã»ã»+nC(n-1) a+1
ã¯ã
ãã(1+1)^n-1^n= nC1 1^(n-1) +nC2 1^(n-2)+nC3 1^(n-3)+ã»ã»ã»ã»+nC(n-1) 1+1
ãã(2+1)^n-2^n= nC1 2^(n-1) +nC2 2^(n-2)+nC3 2^(n-3)+ã»ã»ã»ã»+nC(n-1) 2+1
ãã(3+1)^n-3^n= nC1 3^(n-1) +nC2 3^(n-2)+nC3 3^(n-3)+ã»ã»ã»ã»+nC(n-1) 3+1
ãã(4+1)^n-4^n= nC1 4^(n-1) +nC2 4^(n-2)+nC3 4^(n-3)+ã»ã»ã»ã»+nC(n-1) 4+1
ãã(5+1)^n-5^n= nC1 5^(n-1) +nC2 5^(n-2)+nC3 5^(n-3)+ã»ã»ã»ã»+nC(n-1) 5+1
ããããããããããããããããããããã»
ãã(r+1)^n-r^n= nC1 r^(n-1) +nC2 r^(n-2)+nC3 r^(n-3)+ã»ã»ã»ã»+nC(n-1) r+1
ããããããããããããããããããããã»
+)ã(a+1)^n-a^n= nC1 a^(n-1) +nC2 a^(n-2)+nC3 a^(n-3)+ã»ã»ã»ã»+nC(n-1) a+1
---------------------------------------------------------------------------------
ãã(a+1)^n-1^n=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}+a
ãã(a+1)^n=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}+a+1
ãšããã§ãnãå¥çŽ æ°ãªãã°nCsã¯nã®åæ°ã§ãããã
(a+1)^n=nB+a+1---(6)
ãã ããnB=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}
ããŠã(6)åŒã§ãα=a+1ãšãããšã
α^n=nB+α----(7)
α^n-α=nB
α{α^(n-1)-1}=nB
ãããnBã¯ãn,αã®åæ°ã§ãããã
ãããã£ãŠã
α{α^(n-1)-1}=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}
ãªãã
â =1^(n-1)+2^(n-1)+3^(n-1)+ã»ã»ã»ã»ã»+(a-1)^(n-1)+a^(n-1)
â¡=1^(n-2)+2^(n-2)+3^(n-2)+ã»ã»ã»ã»ã»+(a-1)^(n-2)+a^(n-2)
â¢=1^(n-3)+2^(n-3)+3^(n-3)+ã»ã»ã»ã»ã»+(a-1)^(n-3)+a^(n-3)
â£=1^(n-4)+2^(n-4)+3^(n-4)+ã»ã»ã»ã»ã»+(a-1)^(n-4)+a^(n-4)
ãããããããããããããã»
n-2çªç®=1^2+2^2+3^2+ã»ã»ã»ã»ã»+(a-1)^2+a^2
n-1çªç®=1+2+3+ã»ã»ã»ã»ã»+(a-1)+a
ïŒïŒãã§ã«ããŒã®æçµå®ç
a,b,cãäºãã«çŽ ãªèªç¶æ°ã§ãããnãå¥çŽ æ°ã§ãããªãã°ã
a^n+b^n=c^nãšãããšã(7)åŒããã
nX+a+nY+b=nZ+c
n(X+Y-Z)=c-a-b
X+Y-Z=c/n-a/n-b/n
ããã§ãa,b,cãäºãã«çŽ ãªèªç¶æ°ã§ãããããa,b,cã¯åæã«nã®åæ°ã§ãªãã
ãããã£ãŠã巊蟺ã¯èªç¶æ°ãªã®ã«ãå³èŸºã¯èªç¶æ°ã§ãªãã
ããããc=kn+jãa=ln+jãb=mnãšãããšãc/n=k+j/n,a/n=l+j/n,b/n=mã§å³èŸºã¯èªç¶æ°ã«ãªããããããªãã
ããã§ãa,b,cãäºãã«çŽ ãªèªç¶æ°ã§ãããnã¯å¥çŽ æ°ããã
c (mod n)â b (mod n)
c (mod n)â a (mod n)
a (mod n)â b (mod n)
ããããããªãã
ããã«ããã§ã«ããŒã®æçµå®çã蚌æãããã
No.825ããããã¯ã¡ã¹ã4æ8æ¥ 14:27
以åãšã¯æã£ãŠå€ãã£ãŠã¡ãããšè«çãæžããŠãã ããããã«ãªããã¡ãããšæ°åŠçè°è«ãããã«å€ããèšè¿°ã«ãªã£ãŠããŸããã
> nã¯å¥çŽ æ°ããã
> c (mod n)â b (mod n)
> c (mod n)â a (mod n)
> a (mod n)â b (mod n)
> ããããããªãã
ããã¯ãå³èŸºãèªç¶æ°ã«ãªãããšããããããªãã®æå³ã ãšè§£éããŸããããããã« 2 ã€ããã³ãã©ããããããŸããã
ãŸã 1 ã€ã4â¡11 (mod7) ã®ããã«ãäºãã«çŽ ã§ãå¥çŽ æ°ãæ³ãšããŠååã«ãªãå Žåã¯ããåŸãŸãã
ãã 1 ã€ãããããå¥ã« a, b, c ãååã§ãªããŠã 10/7 - 2/7 - 1/7 = 1 ã®ããã« 3 ã€ã®åæ°ã®åãå·®ãæŽæ°ã«ãªãããšã¯ããåŸãŸãã
No.827DD++4æ8æ¥ 15:19
DD++æ§ãããã«ã¡ã¯ã
ïŒïŒãã§ã«ããŒã®æçµå®ç
ã¯ã
ïŒïŒÎ±^n=nB+αã®å°åº
ã®å¿çšäŸãšããŠãäœã£ããã®ã§ãããè©°ããçãã£ãããã§ããã
äºãã«çŽ ãªèªç¶æ°a,b,c
ãšããã®ããããŠã
c (mod n)â b (mod n)
c (mod n)â a (mod n)
a (mod n)â b (mod n)
ãšãããšããããè¯ãã£ãããªïŒ
ã§ãã
a=jn+gãšããŠa (mod n)â¡g
b=kn+hãšããŠb(mod n)â¡h
c=ln+iãšã㊠c(mod n)â¡i
ãšããŠãc/n-a/n-b/nã§ãi-g-h=0ã¯ããããŸããã
No.828ããããã¯ã¡ã¹ã4æ8æ¥ 16:08
ãã§ã«ããŒã®æçµå®çã¯ãã©ã㪠a, b, c ã§ãããã®çåŒãæç«ããªããšãããã®ã§ãã
èªåã§åæã«æ¡ä»¶ãè¶³ãã a, b, c ã§è°è«ãå§ããŠããŸã£ããããã¯ããæçµå®çãšã¯å¥ã®è©±ã«ãªã£ãŠããŸããŸãã
No.829DD++4æ8æ¥ 16:16
ïŒãšããã§ãnãå¥çŽ æ°ãªãã°nCsã¯nã®åæ°ã§ãããã
ïŒÎ±^n=nB+α----(7)
α^nïŒÎ±ïŒnBããã®å³èŸºãïœã§å²ãåããã®ã§å·ŠèŸºãïœã§å²ãåããã
âŽÎ±^nïŒÎ±â¡ïŒïŒmod ïœïŒããã§ãαãšïœãäºãã«çŽ ãšãããšã䞡蟺ãαã§å²ããã
âŽÎ±^(n-1)ïŒïŒâ¡ïŒïŒmod ïœïŒ
âŽÎ±^(n-1)â¡ïŒïŒmod ïœïŒãã ããïœã¯çŽ æ°ã§Î±ãšïœã¯äºãã«çŽ
https://manabitimes.jp/math/680
No.830KY4æ8æ¥ 16:53
KYæ§ãããã°ãã¯ã
ãªãã»ã©ããã§ã«ããŒã®å°å®çã§ãããæ°ã¥ããŸããã§ããã
No.831ããããã¯ã¡ã¹ã4æ8æ¥ 19:08
ãã§ã«ããŒã®å°å®çãšããã°ããœãã£ãŒã»ãžã§ã«ãã³ã®ãã§ã«ããŒã®æçµå®çãžã®æ¥çžŸã§ããããéã£ãããªïŒ
ç§ãšå£ããæãããåããããªææ³ã§ãã§ã«ããŒã®æçµå®çã蚌æããŠããŸãã
ãã§ã«ããŒã®æçµå®çã®å¶éïŒ a + b ïŒ c ã çŽ æ°ã§ããïŒä»ãã®äžè¬ã® n ã®å Žå
http://y-daisan.private.coocan.jp/html/pdf/felmer-8-1.pdf
ããã«ããããŠã
ãã§ã«ããŒã®æçµå®çã®å¶éïŒ a + b ïŒ c ã n ã®åæ°ã§ãªãïŒä»ãã®äžè¬ã® n ã®å Žå
http://y-daisan.private.coocan.jp/html/pdf/felmer-8-2.pdf
ãšãããŸããã
No.832ããããã¯ã¡ã¹ã4æ8æ¥ 22:44
ïŒãã§ã«ããŒã®å°å®çãšããã°ããœãã£ãŒã»ãžã§ã«ãã³ã®ãã§ã«ããŒã®æçµå®çãžã®æ¥çžŸã§ããããéã£ãããªïŒ
倩æãã§ã«ããŒ
ããã§ã«ããŒã¯ããã€ãã®å®éšçãªèгå¯ããäžè¬çã«æãç«ã€åœé¡ãèŠã€ãã倩æã§ã倧尿§ã
ãªåœé¡ãæ®ããŸãããäŸãã°ãèªç¶æ°ã®ïŒä¹ããïŒãåŒããŸãããããšã
ïŒ^2ïŒïŒïŒïŒïŒïŒÃïŒïŒ(ïŒ^2ïŒïŒïŒïŒïŒïŒïŒ^2ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒ
ïŒ^2ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒ^2ïŒïŒïŒïŒïŒïŒïŒïŒ^2ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒ
ïŒ^2ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒïŒ^2ïŒïŒïŒïŒïŒïŒ
ãªã©ãšãªããŸããå
ã®å®æ°ïœãïŒã®åæ°ãªãã°ããã®ïŒä¹ããïŒãåŒãã°ïŒã®åæ°ã§ãªããªãã®ã¯åœããåã§ãã®ã§ãïŒãïŒå
ã«å
¥ããŸãããããã§ãªããšãã«ã¯ãïœ^2ïŒïŒã¯ïŒã®åæ°ã«ãªã£ãŠããããã§ãããèªç¶æ°ãïŒä¹ããŠïŒãåŒããšã©ããªãã§ãããã
ïŒ^4ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒ^4ïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒ
ïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒ
ïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒïŒ^4ïŒïŒïŒïŒïŒïŒïŒïŒïŒÃïŒïŒïŒïŒ
ãšãªã£ãŠãå
ã®æ°ïœãïŒã®åæ°ã§ãªããã°ãïœ^4ïŒïŒã¯ïŒã®åæ°ã«ãªã£ãŠããããã§ãããã§ã«ããŒã«ãšã£ãŠã¯ãããã¯æ¬¡ã®ããã«äžè¬åããã®ã¯ããããä»äºã§ããïŒ
ãïœãçŽ æ°ïœãšäºãã«çŽ ãªãã°ãïœ^(p-1)ïŒïŒã¯ïœã§å²ãåãã
ãããæ°è«ã«ãããæãåºæ¬çãªå®çããã§ã«ããŒã®å°å®çã§ããèšå·ã§ã¯ïœ^(p-1)â¡ïŒïŒmod ïœïŒãšæžããŸãããšãŠãçŸããå®çã§ãããã
ãæ°åŠã®è±æãäžææ»èãã
å ã¿ã«ã蚌æãããã®ã¯ã©ã€ããããã ããã§ãããŠã£ãããã£ã¢ã«ã¯ïŒéãã®èšŒææ³ãèŒã£ãŠããŸãããäœãåºããã®ã¯åããŠèŠãŸãããæ¯éããã¡ã€ã«ã«ã§ãããŠæ®ããŠäžããã
No.833KY4æ9æ¥ 07:23
KYæ§ããã¯ããããããŸãã
ããããšãããããŸãã
æ©éãPDFã«ããŸããã
ãã§ã«ããŒã®å°å®çã®èšŒæ
http://y-daisan.private.coocan.jp/html/felmer-7-2.pdf
No.834ããããã¯ã¡ã¹ã4æ9æ¥ 09:52
以åãããã ã£ãã®ã§ãããhttp://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã§æ€çŽ¢ããŠã蟿ãçããŸããã§ãããïŒå€åãç§ã®ããœã³ã³ããããããšæãã®ã§ãããïŒ
å ã¿ã«ãã°ãŒã°ã«æ€çŽ¢ã§ã¯ã
https://www.google.com/search?q=http%3A%2F%2Fy-daisan.private.coocan.jp%2Fhtml%2Ffelmer-7-2.pdf&hl=ja&ei=phkyZP7OAdu02roPrfCbsAg&ved=0ahUKEwj-ufa-15v-AhVbmlYBHS34BoYQ4dUDCA8&oq=http%3A%2F%2Fy-daisan.private.coocan.jp%2Fhtml%2Ffelmer-7-2.pdf&gs_lcp=Cgxnd3Mtd2l6LXNlcnAQDEoECEEYAFAAWABgAGgAcAB4AIABAIgBAJIBAJgBAA&sclient=gws-wiz-serp
ã€ããŒæ€çŽ¢ã§ã¯ã
https://search.yahoo.co.jp/search?p=http%3A%2F%2Fy-daisan.private.coocan.jp%2Fhtml%2Ffelmer-7-2.pdf&fr=top_ga1_sa&ei=UTF-8&ts=32910&aq=-1&oq=&at=&ai=8ca69f0b-b347-4657-b8ca-050b0f50e8b7
ãšããç»é¢ã§ããç§ã«ã¯èŠãããŸãããã倧åã«ä¿ç®¡ããŠäžããã
No.835KY4æ9æ¥ 10:57
KY ããã¯ãªã URL ãæ€çŽ¢ããããšæã£ããã ããâŠâŠã
URL ãäœãªã®ãããã£ãŠãªãã®ããšãæããŸãããããã®å²ã«ã¯ Google æ€çŽ¢ã®çµæç»é¢ãèªåã§ URL æžããŠãŸããã
ããŒãïŒïŒ
No.836DD++4æ9æ¥ 11:21
ã§ãPDF èªãŸããŠããã ããŸããã
ãã¡ããšèšŒæã§ããŠãããšæããŸãã
Wikipedia ã®èšŒæ (2) ãšåæ§ã®ãã®ã§ããããã¡ãã¯ããªãçç¥ããæžãæ¹ãããŠããã®ã«å¯ŸããŠãã¯ã¡ã¹ãããã¯ããªãäžå¯§ã«é²ããŠããã£ããããŸããã
现ããç¹ã§ãããæ¹åç¹ã 1 ã€ã
2 ããŒãžç®åŸåãã 3 ããŒãžç®ååã«ãã㊠a = 6 ã§ã®äŸç€ºãããŠãããããã
1^(n-1) + 2^(n-1) + 3^(n-1) + 4^(n-1) + âŠâŠ + 5^(n-1)
ã¿ããã«ãªã£ãŠããŸããã4 ã®æ¬¡ã 5 ãªã®ã§ããã®ãâŠâŠãã¯äžèŠããªãšæããŸãã
No.837DD++4æ9æ¥ 11:33
DD++ãããããã«ã¡ã¯ãç§ã¯ä»¥åãéããããããšãããã³ãã«ããŒã ã䜿ã£ãŠãããã®ã§ããã
ïŒKY ããã¯ãªã URL ãæ€çŽ¢ããããšæã£ããã ããâŠâŠã
ãããããŠãhttp://y-daisan.private.coocan.jp/html/felmer-7-2.pdfããã¯ãªãã¯åºæ¥ãããã«ãªã£ãŠããã®ã§ããããã
ç§ã®ããœã³ã³ã§ã¯ã¯ãªãã¯åºæ¥ãªãã®ã§ãã³ããããŠæ€çŽ¢ããã®ã§ããã蟿ãçããŸããã§ããã
ïŒWikipedia ã®èšŒæ (2) ãšåæ§ã®ãã®ã§ããããã¡ãã¯ããªãçç¥ããæžãæ¹ãããŠããã®ã«å¯ŸããŠãã¯ã¡ã¹ãããã¯ããªãäžå¯§ã«é²ããŠããã£ããããŸããã
èšãããŠèŠçŽããŠã¿ãŸãããã確ãã«çè«çã«ã¯åãã§ããã
No.838KY4æ9æ¥ 12:51
ããããªãã»ã©ãäœã§è©°ãŸã£ãŠããããããŸããã
ãããã web ãã©ãŠã¶ã£ãŠãURL ãæå®ã㊠web ãµã€ããé²èЧããããŒã«ãã§ãã
ãªã³ã¯ãã¯ãªãã¯ãããããã¯ããŒã¯ã§ç§»åãããããã®ã¯ã åã« URL ãæå®ããæéãçããŠãããã ãã§ããæ¬æ¥éãèªåã§çŽæ¥ URL ãæå®ããŠããµã€ããé²èЧã§ããããã§ããã
æå
¥åãé¢åãªãæåãæåŸã®æåãæ¬ èœãããäœèšãªæåãå
¥ã£ããããªãããã«æ³šæããªããã³ããŒããŒã¹ãã§å€§äžå€«ã§ãã
âŠâŠ ã£ãŠããšã§åœãã£ãŠãŸããïŒ
ã¡ãã»ãŒãžäžã® URL ãèªåã§ãªã³ã¯ã«ããæ©èœãç¡å¹ã«ãªã£ãŠããã®ã¯ãããããã¹ãã ãšãã®å¯Ÿçã§ãããã
æ®éã®ãŠãŒã¶ãŒãèªåã§ãªã³ã¯ããŠã»ãããšæã£ããšãã¯ãã¡ãã»ãŒãžæ¬ãããªã URL æ¬ã«ãã® URL ãæžãã°ããããã§ããã
ïŒã¯ã¡ã¹ãããããªãããããªãã£ãã®ãã¯ãç§ã«ã¯ããããŸãããïŒ
No.840DD++4æ9æ¥ 13:27
KYæ§ãããã«ã¡ã¯ã
ïŒã€ããŒæ€çŽ¢ã§ã¯ã
https://search.yahoo.co.jp/search?p=http%3A%2F%2Fy-daisan.private.coocan.jp%2Fhtml%2Ffelmer-7-2.pdf&fr=top_ga1_sa&ei=UTF-8&ts=32910&aq=-1&oq=&at=&ai=8ca69f0b-b347-4657-b8ca-050b0f50e8b7
ãšãªããŸããããã®https://ã»ã»ã»ã»ã»ãšè¡šç€ºãããŠãããšããã«ã
http://y-daisan.private.coocan.jp/html/felmer-7-2.pdf
ã匵ãä»ããã®ã§ããã€ãŸããã³ããŒããŒã¹ãããã®ã§ãã
ä»ãã®ç»é¢ãèŠãŠãããšãã¯ã
(http://)shochandas.xsrv.jp(/ïŒã(ã)ã§å²ãŸããŠããéšåã¯è¡šç€ºãããŸããã
ãšè¡šç€ºãããŠããè¡ã§ããããã«ãhttp://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã³ããŒããŒã¹ãããŸãã
ãã®ç»é¢ããæ»ã£ãŠããã«ã¯ããâïŒå·Šç¢å°ïŒããã¯ãªãã¯ãããšãã®ç»é¢ã«æ»ããŸãã
No.841ããããã¯ã¡ã¹ã4æ9æ¥ 13:34
ããœã³ã³çŽ äººãªè
ã§ãã¿ãŸããã
ïŒæå
¥åãé¢åãªãæåãæåŸã®æåãæ¬ èœãããäœèšãªæåãå
¥ã£ããããªãããã«æ³šæããªããã³ããŒããŒã¹ãã§å€§äžå€«ã§ãã
http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfããã®ãŸãŸã³ããŒããŒã¹ãããŠã°ãŒã°ã«æ€çŽ¢ããŠãã€ããŒæ€çŽ¢ããŠããã®URLã«è¡ããªããã§ããæ®éã¯ããã€ãäžçªäžã«åºãã®ã§ãããããããã¯ã¡ã¹ãããã®äŸãã°ã
ãã§ã«ããŒã®æçµå®çã®å¶éïŒ a + b ïŒ c ã çŽ æ°ã§ããïŒä»ãã®äžè¬ã® n ã®å Žå
http://y-daisan.private.coocan.jp/html/pdf/felmer-8-1.pdf
ã§ïŒã°ãŒã°ã«ïŒæ€çŽ¢ãããšäžçªäžã«ã奿°ã®å®å
šæ°ã¯ãªãããåºãŸããããã§ããhttp://y-daisan.private.coocan.jp/html/kanzensu.pdf
ïŒã¡ãã»ãŒãžäžã® URL ãèªåã§ãªã³ã¯ã«ããæ©èœãç¡å¹ã«ãªã£ãŠããã®ã¯ãããããã¹ãã ãšãã®å¯Ÿçã§ãããã
æ®éã®ãŠãŒã¶ãŒãèªåã§ãªã³ã¯ããŠã»ãããšæã£ããšãã¯ãã¡ãã»ãŒãžæ¬ãããªã URL æ¬ã«ãã® URL ãæžãã°ããããã§ããã
ïŒã¯ã¡ã¹ãããããªãããããªãã£ãã®ãã¯ãç§ã«ã¯ããããŸãããïŒ
ãããããŠé ããã°èŸ¿ãçãããšæããŸããã
No.842KY4æ9æ¥ 14:47
URLãèšå
¥ããŸãããããã§ã©ãã§ãããïŒ
ç·è²ã®ãããããã¯ã¡ã¹ãããã¯ãªãã¯ããŠãã ããã
ããããæ©èœã䜿ã£ãŠãªãã£ãããã§ããŠã£ãããèªåã®ããŒã ããŒãžã貌ãããã ãšæã£ãŠããŸããã
ãã ãæ°ã«ãªãããšããããŸããTexã§PDFãã€ãããšããã©ã³ããåã蟌ãŸããªãã®ã§ãWindowsã§ã¯ãæ¥æ¬èªã®æåãå€ãªæåã«å²ãæ¯ãããããšããããŸãã
ã¢ã¯ãããããªãŒããŒãã€ã³ã¹ããŒã«ããŠããã°ãåé¡ãªãã®ã§ããã
ãã¡ãããã¢ã¯ãããããªãŒããŒã¯ãç¡æã§ãã
ããããã¯ã¡ã¹ããããããããšãããããŸããç¡äºã蟿ãçããŸããã
DD++ããã®ææã®ã
ïŒçްããç¹ã§ãããæ¹åç¹ã 1 ã€ã
2 ããŒãžç®åŸåãã 3 ããŒãžç®ååã«ãã㊠a = 6 ã§ã®äŸç€ºãããŠãããããã
1^(n-1) + 2^(n-1) + 3^(n-1) + 4^(n-1) + âŠâŠ + 5^(n-1)
ã¿ããã«ãªã£ãŠããŸããã4 ã®æ¬¡ã 5 ãªã®ã§ããã®ãâŠâŠãã¯äžèŠããªãšæããŸãã
ããã¯çŽãããã®ã§ããããã倧äžå€«ã¿ããã§ããã
ïŒWikipedia ã®èšŒæ (2) ãšåæ§ã®ãã®ã§ããããã¡ãã¯ããªãçç¥ããæžãæ¹ãããŠããã®ã«å¯ŸããŠãã¯ã¡ã¹ãããã¯ããªãäžå¯§ã«é²ããŠããã£ããããŸããã
ïœ^pïŒ[ïŒïŒ(ïœïŒïŒ)]^p
ããâ¡ïŒ^pïŒ[ïŒïŒ(ïœïŒïŒ)]^p
ããâ¡ïŒïŒïŒ^pïŒ[ïŒïŒ(ïœïŒïŒ)]^p
ããâ¡ïœïŒmod ïœïŒ
ãã¡ãã®æ¹ããã»ã»ã»ãå
¥ããæ¹ãè¯ããšæããŸãã
å ã¿ã«ããããèŠãŠããããã¯ã¡ã¹ãããã®è§£æ³ãäœãã人ã¯ããŸãããã
No.844KY4æ9æ¥ 16:52
ãããã ãããªãæ€çŽ¢æ¬ã«æã¡èŸŒãã®ãâŠâŠã
æ€çŽ¢æ¬ã¯ãããã«é¢ä¿ããæ
å ±ãããå ŽæãæããŠãã ãããã§ãã
ããã®å Žæã«é£ããŠè¡ã£ãŠãã ãããã§ã¯ãããŸãããããããªæ©èœã¯ Google ã«ã Yahoo ã«ããããŸããã
No.845DD++4æ9æ¥ 17:00
ããããæå³ãåãããŸãããçŽæ¥äžã®éµå°ã®æã«ã³ããããã°è¯ãã£ãã®ã§ããã
ãéšããããŸããã
No.846KY4æ9æ¥ 18:57
管ç人ãããšã¯å¥ã®ããæ¹ã§æ±ããŠã¿ãŸããã
α = cos(2Ï/7), β = cos(4Ï/7) , γ = cos(6Ï/7) ãšããŸãã
åè§ã®å
¬åŒãã (cotx)^2 = (1+cos2x)/(1-cos2x) ãªã®ã§ã
(1+α)/(1-α) + (1+β)/(1-β) + (1-γ)/(1+γ) ãæ±ããã°ããããšã«ãªããŸãã
ζ = exp(2Ï/7) = cos(2Ï/7) + i sin(2Ï/7) ãšãããŸãã
z^7 - 1 = 0 㯠1 ã® 7 乿 ¹ãè§£ã«ãã€ã®ã§ã
z^7 - 1 = ( z - 1 ) ( z - ζ ) ( z - ζ^2 ) ( z - ζ^3 ) ( z - ζ^4 ) ( z - ζ^5 ) ( z - ζ^6 )
= ( z - 1 ) { ( z - ζ ) ( z - ζ^6 ) } { ( z - ζ^2 ) ( z - ζ^5 ) } { ( z - ζ^3 ) ( z - ζ^4 ) }
= ( z - 1 ) ( z^2 - 2αz + 1 ) ( z^2 - 2βz + 1 ) ( z^2 - 2γz + 1 )
ããã ( z - 1 ) z^3 ã§å²ããš
( z^3 + 1/z^3 ) + ( z^2 + 1/z^2 ) + ( z + 1/z ) + 1 = ( z + 1/z - 2α ) ( z + 1/z - 2β ) ( z + 1/z - 2γ )
ããã§ã2x = z + 1/z ãšãããšã
4x^2 = z^2 + 2 + 1/z^2 ãããz^2 + 1/z^2 = 4x^2 - 2
8x^3 = z^3 + 3z + 3/z + 1/z^3 ãããz^3 + 1/z^3 = 8x^3 - 6x
ãªã®ã§ããããçšããŠæžãæãããš
8x^3 + 4x^2 - 4x - 1 = 8 ( x - α ) ( x - β ) ( x - γ )
t = (1+x)/(1-x) ãšãããšãx = (t-1)/(t+1) ãªã®ã§ãããã代å
¥ããŠäž¡èŸºã« (t+1)^3 ãããããš
8(t-1)^3 + 4(t-1)^2*(t+1) - 4(t-1)(t+1)^2 - (t+1)^3 = { (t-1) - (t+1)α } { (t-1) - (t+1)β } { (t-1) - (t+1)γ }
æŽçããŠ
7t^3 - 35t^2 + 21t - 1 = { (1-α)t - (1+α) } { (1-β)t - (1+β) } { (1-γ)t - (1+γ) }
ãã£ãŠ (1+α)/(1-α), (1+β)/(1-β) ,(1-γ)/(1+γ) ã¯æ¹çšåŒ 7t^3 - 35t^2 + 21t - 1 = 0 ã®è§£ãªã®ã§ã
è§£ãšä¿æ°ã®é¢ä¿ãããã®å㯠35/7 = 5
No.785DD++4æ2æ¥ 07:27
æçš¿åŸã«ã7t^3 - 35t^2 + 21t - 1 ãšããã©ãèŠãŠã 7Ck ãªä¿æ°ãèŠãŠã
å
ã»ã©ã®ã¯ãšãã§ããªãé åããããŠããããšã«æ°ã¥ããŠããŸã£ãâŠâŠã
kÏ/7 㯠7 åãããš Ï ã®æŽæ°åã«ãªãã®ã§ã
cot(kÏ/7) + i = { cos(kÏ/7) + i sin(kÏ/7) } / sin(kÏ/7) 㯠7 ä¹ãããšå®æ°ã§ãã
ãã£ãŠã6 次æ¹çšåŒ (x+i)^7 - (x-i)^7 = 0ã®è§£ã¯ x = ±cot(Ï/7), ±cot(2Ï/7), ±cot(3Ï/7) ã§ããã¯ãã§ãã
ãã®æ¹çšåŒã®å·ŠèŸºãå
šéšå±éãããšãã
x^6 ã®ä¿æ°ã¯ 2*7C1*i^1
x^4 ã®ä¿æ°ã¯ 2*7C3*i^3
ãã®æ¯ 7C3/7C1*i^2 = -35/7 = -5 ã¯ã6 ã€ã®è§£ã®ç°ãªã 2 ã€ãã€ã®ç©ã®ç·åã§ããã
笊å·éããæã¡æ¶ãåãããšãèããã°ããã㯠- {cot(Ï/7)}^2 - {cot(2Ï/7)}2 - {cot(3Ï/7)}^2 ã«ä»ãªããŸããã
ãã£ãŠã{cot(Ï/7)}^2 + {cot(2Ï/7)}2 + {cot(3Ï/7)}^2 = 5
No.786DD++4æ2æ¥ 07:51
ãšããã§ã2åè§ã3åè§ã®å
¬åŒã¯
ããããã2tanΞ
tan2Ξ=-----------
ãããã1-(tanΞ)^2
ãããã1-(cotΞ)^2
cot2Ξ=-----------
ããããã2cotΞ
ããã¯ããªããšãªãéæ°ãšããã€ã¡ãŒãžãªã®ã§ããã£ããããªæ°ã«ãªããã§ãããïŒã§ãæ°åŠçã«ãããããšããå°è±¡ã§ãããïŒïŒ
ãããã
ãããã3tanΞ-(tanΞ)^3
tan3Ξ=-----------------
ãããã1-3(tanΞ)^2
ãããã(cotΞ)^3-3cotΞ
cot3Ξ=------------------
ããããã3(cotΞ)^2-1
ãšå
šãåãæ§é ãªãã§ãããäžæè°ã§ãããïŒã§ãæ°åŠçã«ãããããšããå°è±¡ã§ãããïŒïŒ
No.787ããããã¯ã¡ã¹ã4æ3æ¥ 20:20
æ°åŠãã¡ãããšãã人ã¯ãããããæ°åŠçã«ãããããã©ãããå°è±¡ã§èšã£ããã¯çµ¶å¯Ÿã«ããŸããã
No.789DD++4æ4æ¥ 00:15
åãééããŠãŸããã
ãããã(cotΞ)^2-1
cot2Ξ=-----------
ããããã2cotΞ
ã§ããããããããå¶æ°ãšå¥æ°ã§éã«ãªãã®ãããããŸããããïŒç ç©¶ããŠäžãããïŒ
No.790éãããã4æ4æ¥ 07:17
DD++æ§ããã¯ããããããŸãã
(%i1) float((cot(%pi/7))^2+(cot(2*%pi/7))^2+(cot(3*%pi/7))^2);
(%o1) 5.000000000000001
ãªãã¯ãtanãšcotã®å
ã®å°è±¡ããã
(%i2) float((tan(%pi/7))^2+(tan(2*%pi/7))^2+(tan(3*%pi/7))^2);
(%o2) 20.99999999999999
ããã21ã«ãªããããªå°è±¡ã§ããã»ã»ã»ã»
éããããæ§ããã¯ããããããŸãã
åãééããŠããŸããããææãããããšãããããŸãã
No.791ããããã¯ã¡ã¹ã4æ4æ¥ 07:18
tan ã®æ¹ã¯ã7 次æ¹çšåŒ (1+xi)^7 - (1-xi)^7 = 0 ã®è§£ã x = 0, ±tan(Ï/7), ±tan(2Ï/7), ±tan(3Ï/7) ã§ããããšããã7C2/7C0 ç±æ¥ã§ 21 ãåŸãããŸããã
No.793DD++4æ4æ¥ 10:29
ã¡ãã£ãšãåé¡ããå€ããŸãããïŒïŒãïŒïŒãäžããŸãã
ïŒïŒ
ãããã(cotΞ)^2-1
cot2Ξ=-----------------
ããããã2cotΞ
ãããã(cotΞ)^3-3cotΞ
cot3Ξ=-----------------------
ããããã3(cotΞ)^2-1
ã䜿ã£ãŠã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2=5ãããã£ãŠã¿ãã
cotΞ=xãšãããšã
ããããx^2-1ãããããx^3-3x
x^2ïŒ(------------)^2ïŒ(-----------------)^2=5ãã
ããããã2xãããããã3x^2-1
49x^8-72x^6+62x^4-8x^2+1
----------------------------------------=5
ããã4x^2(3x^2-1)^2
ããã
49x^8-72x^6+62x^4-8x^2+1=20x^2(3x^2-1)^2
(7x^2-1)(7x^6-35x^4+21x^2-1)=0
ããã7x^2-1=0ãš7x^6-35x^4+21x^2-1=0ããæãç«ãŠã°ã
(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2=5ãã¯ããã ããã
7x^2-1=0ã§ã¯x=±1/â7ã宿°è§£ãããã
7x^6-35x^4+21x^2-1=0ã§ã¯ã宿°è§£ã¯ãªããè€çŽ æ°è§£ã¯ããã
ããã«ã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2=5ãã¯ã宿°è§£ãããã®ã§ãã ããã
ïŒïŒ
ããããã2tanΞ
tan2Ξ=----------------
ãããã1-(tanΞ)^2
ãããã3tanΞ-(tanΞ)^3
tan3Ξ=------------------------
ãããã1-3(tanΞ)^2
ã䜿ã£ãŠã(tanΞ)^2+(tan2Ξ)^2+(tan3Ξ)^2=21ãããã£ãŠã¿ãã
tanΞ=xãšãããšã
ãããã2xãããããã3x-x^3
x^2ïŒ(------------)^2ïŒ(---------------)^2=21ãã
ãããã1-x^2ããããã1-3x^2
2x^2(5x^8-16x^6+40x^4-28x^2+7)
------------------------------------------------=21
ãã(x-1)^2(x+1)^2(3x^2-1)^2
2x^2(5x^8-16x^6+40x^4-28x^2+7)=21(x-1)^2(x+1)^2(3x^2-1)^2
(2x^2-1)(5x^2-3)(x^6-21x^4+35x^2-7)=0
ããã2x^2-1=0ã5x^2-3=0ãšx^6-21x^4+35x^2-7=0ããæãç«ãŠã°ã
(tanΞ)^2+(tan2Ξ)^2+(tan3Ξ)^2=21ãã¯ããã ããã
2x^2-1=0ã§ã¯x=±1/â2ã宿°è§£ãããã
5x^2-3=0ã§ã¯x=±â3/â5ã宿°è§£ãããã
x^6-21x^4+35x^2-7=0ã§ã¯ã宿°è§£ã¯ãªããè€çŽ æ°è§£ã¯ããã
ããã«ã(tanΞ)^2+(tan2Ξ)^2+(tan3Ξ)^2=21ã¯ã宿°è§£ãããã®ã§ãã ããã
No.794ããããã¯ã¡ã¹ã4æ4æ¥ 12:52
ïŒ7x^6-35x^4+21x^2-1=0ã§ã¯ã宿°è§£ã¯ãªããè€çŽ æ°è§£ã¯ããã
宿°è§£ãïŒåãããŸãããïœïŒÂ±0.2282432,±0.7974733,±2.0765214
æåŸã®ã§ÎžïŒÏ/7ãåããšæããŸãã
No.795éãããã4æ4æ¥ 16:04
éããããæ§ãããã°ãã¯ã
倧å€ãããããšãããããŸãïŒ
ãã£ãšãã±ãªãã€ããŸããã
(%i12) fpprec:50; 50æ¡æå®
(%o12) 50
(%i13) x:bfloat(cot(%pi/7));
(%o13) 2.076521396572336567163538861485840330705720206626b0
(%i14) 7*x^6-35*x^4+21*x^2-1;ã«ä»£å
¥
(%o14) - 6.8422776578360208541197733559077936097669040130689b-49
çã
ã»ãŒïŒã§ãã
è¿äŒŒè§£ãæ±ãããšã
(%i1) allroots( 7*x^6-35*x^4+21*x^2-1);
(%o1) [x = 0.2282434743901499, x = - 0.2282434743901499,
x = 0.7974733888824038, x = - 0.797473388882404, x = - 2.076521396572337,
x = 2.076521396572336]
ãšéããããæ§ã®çµæã«ãªããŸãã
ãŸããtanïŒÏ/7ïŒã¯ã
float(tan(%pi/7));
(%o3) 0.4815746188075286
ã§ã
è¿äŒŒè§£ãæ±ãããšã
(%i2) allroots(x^6-21*x^4+35*x^2-7);
(%o2) [x = - 0.4815746188075286, x = 0.4815746188075286,
x = - 1.253960337662704, x = 1.253960337662703, x = 4.381286267534823,
x = - 4.381286267534823]
ãšãªããtanïŒÏ/7ïŒããããŸããã
No.796ããããã¯ã¡ã¹ã4æ4æ¥ 16:55
No.794 ã®èšäºã ãå
šç¶éãåé¡ã«åãã£ãŠããã®ã¯æå³çã«ãã£ãŠãããã®ãªãã§ããããïŒ
ãããŠæå³çãªã®ã ãšããããåãã£ãŠããåé¡ãè¿°ã¹ãŠããå§ããŠãã ããã
ãªãã4åã»ã©ããã ããããšèšã£ãŠããŸããããäœããã ãããã®ã誰ã«ãããããŸããã
No.797DD++4æ4æ¥ 18:22
DDïŒïŒæ§ãããã°ãã¯ã
ããšããšã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2=5ã蚌æãããã ã£ãã®ã§ããã£ã¡ã«äž»çŒã眮ããŸãããΞïŒÏïŒïŒãèŠããªãã£ãããã§ãã
(tanΞ)^2+(tan2Ξ)^2+(tan3Ξ)^2=21ãããã§ãã
ã§ããéããããæ§ã®ç ç©¶çµæãããΞïŒÏïŒïŒãèŠããŠããã®ã§ãã
ããã§ãæµãããããã颚ã«ãªã£ãã®ã§ãããã¿ãŸããã
No.798ããããã¯ã¡ã¹ã4æ4æ¥ 18:36
ããã§ããããΞã Ï/7 ã«éããªã話ãããŠããŸãããã
ã ãšããããããã ããããšã¯äœã®ããšãèšã£ãŠããã®ã§ããïŒ
Ξãå®ãŸã£ãŠããªããªãã°ãΞã®å€ã«ãã£ãŠçåŒã¯æãç«ã£ããæãç«ããªãã£ããããã¯ãã§ããã
No.799DD++4æ4æ¥ 21:49
DD++æ§ããã¯ããããããŸãã
ïŒã ãšããããããã ããããšã¯äœã®ããšãèšã£ãŠããã®ã§ããïŒ
(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2=5
(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2ãŒ5ïŒïŒ
ãæãç«ã€ããšããããšã§ãã
åŒãå±éæŽçãããã
(7x^2-1)(7x^6-35x^4+21x^2-1)=0
ãšãªã£ãã®ã§ã7x^2-1=0ãããããã¯7x^6-35x^4+21x^2-1=0ãšãªããããããã
(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2ãŒ5ïŒïŒ
ã¯ãΞã«ãã£ãŠã¯ãæãç«ã€ã®ã§ãæ£ãããšèšã£ãŠããã®ã§ãããã¡ãããäžçåŒã«ã¯ãªããªãã£ãã§ãããã
ïŒÎžãå®ãŸã£ãŠããªããªãã°ãΞã®å€ã«ãã£ãŠçåŒã¯æãç«ã£ããæãç«ããªãã£ããããã¯ãã§ããã
ããã§ããããã¯ã8ã€ã®è§£ãããã®ã§ãΞã¯ã8éããããŸãã
No.800ããããã¯ã¡ã¹ã4æ5æ¥ 07:28
調ã¹ãŠããŸããããïœïŒÂ±0.2282432,±0.7974733,±2.0765214ã®æ®ãïŒã€ã§ïŒÏ/ïŒãšïŒÏ/ïŒã«å¯Ÿå¿ããŠããã®ã§ã¯ãªãã§ãããããïŒèª¿ã¹ãŠé ãããšçŽåŸåºæ¥ãŸããïŒ
No.801éãããã4æ5æ¥ 07:58
éããããæ§ããã¯ããããããŸãã
maximaã§ã
%i1) solve(x^3+1=0,x);ããããããããããããããããããïœè§£ãæ±ããïœ
ããããããããsqrt(3) %i - 1ãã sqrt(3) %i + 1ããããããã{%iã¯èæ°ïœ
(%o1) [x = - -----------------, x =--------------------, x = - 1]
ãããããããããã2ãããããããã2
(%i2) allroots(x^3+1=0);
(%o2) [x = 0.8660254037844386 %i + 0.5, x = 0.5 - 0.8660254037844386 %i,
x = - 1.0]
ãšãªããŸãã®ã§ãéããããæ§ã¯ã宿°è§£ãæ±ããŠããã®ã§ãããè¿äŒŒè§£ãšã¯ãã¿ãŸããã§ããã
ããŠããæšå¯ã®ãšããã
(%i5) float(cot(%pi/7));
(%o5) 2.076521396572337
(%i6) float(cot(2*%pi/7));
(%o6) 0.797473388882404
(%i7) float(cot(3*%pi/7));
(%o7) 0.22824347439015
Ï/7ãïŒÏ/7ãïŒÏ/7ã§ããããããã§ãã
No.802ããããã¯ã¡ã¹ã4æ5æ¥ 08:02
ã ãšãããã
ã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2ãŒ5ïŒïŒ ã¯ç¹æ®ãªÎžã®ãšãã«æãç«ã€å Žåãããããããã ããããšãã¡ããšè¿°ã¹ãŠãã ããã
æã
ã¯è¶
èœåè
ãããªãã®ã§ãã¯ã¡ã¹ãããã®é ã®äžã«ããååšããªãæã¯èªããŸããã
No.803DD++4æ5æ¥ 08:02
ïŒã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2ãŒ5ïŒïŒ ã¯ç¹æ®ãªÎžã®ãšãã«æãç«ã€å Žåãããããããã ããããšãã¡ããšè¿°ã¹ãŠãã ããã
ããããäžçåŒã«ãªãã°ãæãç«ã€å ŽåããããŸããã®ã§ãæ¹çšåŒãšããŠãæ£ãããããŸãããã
æãç«ã€å Žåãããã®ã§ããã°ãæ¹çšåŒãšããŠããã ããããšèšã£ããçŽåŸããŠããã ããŸããïŒ
No.804ããããã¯ã¡ã¹ã4æ5æ¥ 08:15
æ¹çšåŒãšããŠæ£ãããšã¯ãéåžžããã®åŒãæ¹çšåŒã®å®çŸ©ã«è©²åœãããšããæå³ã§ãã
ããªãã¡ãåŒïŒå®çŸ©ãããèšç®èšå·ãæ£ãã䜿ãããŠããæ°åãšèšå·åïŒãçå·ã®äž¡èŸºã«æžããŠãããããã«æªç¥æ°ãå«ãŸããŠãããã®ã§ããããšããããšã§ãã
ã ããã(cotΞ)^2+(cot2Ξ)^2+(cot3Ξ)^2-5=0ããšæžãããã ãã§ããæ¹çšåŒã®å®çŸ©ã«è©²åœããŠããã®ã ãããæ¹çšåŒãšããŠæ£ããããšã¿ããªèªããŸããã
念ã®ããèšã£ãŠãããšãæ¹çšåŒã®å®çŸ©ã«ãå®éã«æãç«ã€å Žåããããã©ããã¯èšåãããŠããŸããã
è§£ããªãæ¹çšåŒã¯ããã è§£ããªããšããç¹åŸŽããããšããã ãã®æ£ããæ¹çšåŒã§ãã
ã ãããã¯ã¡ã¹ããããããã ããããšèšã£ãŠããå
容ã¯ããããããæ¹çšåŒãšããŠæ£ããããããªããç¹æ®ãªÎžã®ãšãã«æãç«ã€å Žåãããããšæžãããã¹ããã®ãããªãããšæãã®ã§ããã
No.805DD++4æ5æ¥ 10:31
DDïŒïŒæ§ãããã«ã¡ã¯ã
ããããšãããããŸããããææã¯ããããŸããã
No.808ããããã¯ã¡ã¹ã4æ5æ¥ 16:21