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(2) t^3+3*t+1=3*t+3ããªã®ã§ã (3*t+3)*(t^2ïŒtïŒ1)ïŒ9ããããQ=(t^2ïŒtïŒ1)/9ãã§ããïŒ
No.1418管çè
2023幎9æ15æ¥ 16:17 ãã£ããªã解決ããã¡ãããŸããã
No.1420GAI2023幎9æ15æ¥ 20:58
(1)x^2+y^2+sin(7*x)+sin(7*y)-1=0
(2)x^2+y^2+sin(7*x)+cos(7*y)-1=0
(3)x^2+y^2+cos(7*x)+cos(7*y)-1=0
(4)|x|+|y|+sin(|7*x|)+sin(|7*y|)-1=0
(5)|x|+|y|+sin(|7*x|)+cos(|7*y|)-1=0
(6)|x|*|y|+sin(|7*x|)*cos(|7*y|)-1=0
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=(X^2+3*Y^2)*((X^2+3*Y^2)^2-(3*X*Y)^2)
=(X^2+3*Y^2)*(x^2-3*X*Y+3*Y^2)*(X^2+3*X*Y+3*Y^2)
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7*{ (X^15-Y^15) -X^3*Y^3*(X^9-Y^9) +X^6*Y^6*(X^3-Y^3)} +
21*{(X^12+Y^12) -X^3*Y^3*(X^6+Y^6) +(X^3-Y^3)+X^6*Y^6} +
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=(7*X^2+Y^2)
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ã§ã¯ãã®äžã§çµåæ³åãæºããååã¯äœéããããïŒ
(2)Mã®å
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šéšã§3^(3^2)=19683(éã)ã®ååã®äžã§
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šéšã§4^(4^2)=4294967296(éã)ã®ååã®äžã§
çµåæ³åãæºããååã¯äœéããããïŒ
No.1382GAI2023幎8æ20æ¥ 06:37
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No.1383HP管çè
2023幎8æ20æ¥ 21:59 ããæ¬ãèªãã§ãããšãããã®ååãšçµåæ³åã®çµåãã«ã€ããŠã®èšè¿°ãèªãã§
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ã§ããã®ãµã€ãã§ã®èª¬ææã§ã¯äœãçµåæ³åãªãèšè¿°ã¯ãªããæ°ã¯äžèŽããããããæ±ããæ°ã
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No.1384GAI2023幎8æ21æ¥ 20:20
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associative : ãæŒç®ãªã©ããçµåçãªãçµååŸïŒ»æ³åãæºãã
binary operation : äºé
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ãããã®ã§ã
" Number of associative binary operations on an n-set "
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ãèŠçŽ æ°nã®éåã«ãããçµåæ³åãæºããäºé
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No.1385ããã²ã2023幎8æ21æ¥ 23:34
ååãŸã§ã¯ãåææ°ã§ã¯ãèšç®äžã§ã¯æ£ããã®ã§ãããçå±ã§ã¯ãå³èŸºãšå·ŠèŸºã®åæ°ãçãããªããŸããã§ããã
ããã§ã解決ã§ããŸããã
çæ¯çŽæ°ã®åã®å
¬åŒãã
N^s-1=(N-1){1+N+N^2+n^3+ã»ã»ã»ã»+N^(s-2)+N^(s-1)}------(1)
ããŠãs=4ã®ãšãã
N(N-1)(N-2)(N-3)=(N^4-N)-6(N^3-N)+11(N^2-N)-----(2)
=N(N^3-1)-6N(N^2-1)+11N(N-1)
ã§ããããã(1)ããã
=N(N^3-1)-6N(N^2-1)+11N(N-1)
=N{(N-1)(1+N+N^2)-6(N-1)(1+N)+11(N-1)}
=N(N-1){(1+N+N^2)-6(1+N)+11}
=N(N-1){(1+N+N^2-6-6N+11}
=N(N-1){(N^2-5N+6}
=N(N-1)(N-2)(N-3)
ããããã(2)ã®å³èŸºã¯å·ŠèŸºã«çããã
ããŠãs=6ã®ãšãã
N(N-1)(N-2)(N-3)(N-4)(N-5)=(N^6-N)-15(N^5-N)+85(N^4-N)-225(N^3-N)+274(N^2-N)------(3)
ã§ããããã(1)ããã
=N(N^5-1)-15N(N^4-1)+85N(N^3-1)-225N(N^2-1)+274N(N-1)
=N(N-1){(1+N+N^2+N^3+N^4)-15(1+N+N^2+N^3)+85(1+N+N^2)-225(N+1)+274}
=N^4-14N^3+71N^2-154N+120
=N(N-1)(N-2)(N-3)(N-4)(N-5)
ããããã(3)ã®å³èŸºã¯å·ŠèŸºã«çããã
ããŠãs=8ã®ãšãã
N(N-1)(N-2)(N-3)(N-4)(N-5)(N-6)(N-7)=(N^8-N)-28(N^7-N)+322(N^6-N)-1960(N^5-N)+6769(N^4-N)-13132(N^3-N)+13068(N^2-N)------(4)
ã§ããããã(1)ããã
=N{(N^7-1)-28(N^6-1)+322(N^5-1)-1960(N^4-1)+6769(N^3-1)-13132(N^2-1)+13068(N-1)}
=N(N-1){(1+N+N^2+N^3+N^4+N^5+N^6)-28(1+N+N^2+N^3+N^4+N^5)+322(1+N+N^2+N^3+N^4)-1960(1+N+N^2+N^3)+6769(1+N+N^2)-13132(1+N)+13068}
=N^6-27N^5+295N^4-1665N^3+5104N^2-8028N+5040
=N(N-1)(N-2)(N-3)(N-4)(N-5)(N-6)(N-7)
ããããã(4)ã®å³èŸºã¯å·ŠèŸºã«çããã
ãã£ãŠSãèªç¶æ°ã®ãšã
N(N-1)(N-2)(N-3)ã»ã»ã»ã»{N-(S-1)}=(N^S-N)-a1{N^(S-1)-N}+a2{N^(S-2)-N}ã»ã»ã»ã»-a(s-2)(N^3-N)+a(s-1)(N^2-N)
ã¯ãsãåææ°ã§ãæãç«ã€ã
No.1375ããããã¯ã¡ã¹ã2023幎8æ19æ¥ 08:32
ã€ãŸãããªãçå±ãééãããšãããšãå³èŸºã¯ãsãåææ°ã®å Žå(N^t-N){ãã ããtã¯s以äžã®ãã¹ãŠã®èªç¶æ°}ã®ä¿æ°ãsã®åæ°ã«ãªããªããšããã®ããååã®çµè«ã§ããã
ãã ãã(N^t-N)ãtã®åæ°ãšããããæãç«ã€ããã§ãã
(%i1) factor(4-6*3+11*2);
(%o1) 2^3=4X2
(%i2) factor(6-15*5+85*4-225*3+274*2);
%o2) 2^4 3^2=6^2x2^2
(%i3) factor(8-28*7+322*6-1960*5+6769*4-13132*3+13068*2);
(%o3) 2^7 3^3 5=8x2^4x3^3x5
ç·šéæžã¿
No.1376ããããã¯ã¡ã¹ã2023幎8æ19æ¥ 08:50
ããããã¯ã¡ã¹ãããããã¯ããããããŸãã
ïŒãã£ãŠSãèªç¶æ°ã®ãšã
N(N-1)(N-2)(N-3)ã»ã»ã»ã»{N-(S-1)}=(N^S-N)-a1{N^(S-1)-N}+a2{N^(S-2)-N}ã»ã»ã»ã»-a(s-2)(N^3-N)+a(s-1)(N^2-N)
ã¯ãsãåææ°ã§ãæãç«ã€ã
ããã¯ååèªåã§èšŒæãããŸãããããNo.1340ã®æçš¿ã§ãã
ãå ã¿ã«ãäœæ¬¡ã§ã-abcdN+N-(a+b+c+d)N+(ab+bc+cd+ac+ad+bd)N-(abc+abd+acd+bcd)Nã®éšåã¯Nã®é
ã«ãªãã解ãšä¿æ°ã®é¢ä¿ãšåãã§Â±ã亀äºã«ãªããå¿
ã ±(a - 1) (b - 1) (c - 1) (d - 1)âŠãšå æ°å解ã§ããïœïŒïŒããã
N(N-1)(N-2)(N-3)ã»ã»ã»ã»{N-(S-1)}=(N^S-N)-a1{N^(S-1)-N}+a2{N^(S-2)-N}ã»ã»ã»ã»-a(s-2)(N^3-N)+a(s-1)(N^2-N)ã®åœ¢ã«åºæ¥ãã®ã§ããããïŒNo.1342ããïŒ
ãã®ïŒ³ã«ã¯åææ°ãšãçŽ æ°ãšãå¶éããªãã®ã§ãäžè¬ã®ïŒ³ã§æãç«ã€ãšããäºã§ããã
ïŒã€ãŸãããªãçå±ãééãããšãããšãå³èŸºã¯ãsãåææ°ã®å Žå(N^t-N){ãã ããtã¯s以äžã®ãã¹ãŠã®èªç¶æ°}ã®ä¿æ°ãsã®åæ°ã«ãªããªããšããã®ããååã®çµè«ã§ããã
çå±ã¯ééã£ãŠããªããšæããŸããäžå¿ãååã®ãã®ãæããŠãããŸããã
ïŒçåŒã¯æãç«ã€ã®ã§ã巊蟺ã®åæ°ãšå³èŸºã®åæ°ã¯çããã®ã§ãã
ãšããããåææ°ã®ãšãã巊蟺ãšå³èŸºãäžèŽããªããšããçå±ãããããã®ã§ãã
ãã®å³èŸºãÃã ãã§ã€ãªãã£ãåŒãªãããããã§ããã(N^4-N)-6(N^3-N)+11(N^2-N)ã¯åãšå·®ã§ã€ãªãã£ãŠããã®ã§ãããããããŸããã確ããNHKã®çªçµã§ãæãç®ã¯ç°¡åã§ãã足ãç®ã¯é£ãããšãããããªè©±ããã£ãŠããŸããããããããšåãäºã§ãã
ïŒãã ãã(N^t-N)ãtã®åæ°ãšããããæãç«ã€ããã§ãã
(%i1) factor(4-6*3+11*2);
(%o1) 2^3=4X2
(%i2) factor(6-15*5+85*4-225*3+274*2);
%o2) 2^4 3^2=6^2x2^2
(%i3) factor(8-28*7+322*6-1960*5+6769*4-13132*3+13068*2);
(%o3) 2^7 3^3 5=8x2^4x3^3x5
ããã¯ãDD++ãããçºèŠãããN^561-Nã¯ïŒïŒïŒã®åæ°ãšããåææ°561ã§ãã£ãŠã¿ãŠäžãããå€åããã¡ã§ãããïŒå€åã®æã¯èšŒæããŠãããŸãããä»åã¯çç¥ããŸããïŒ
No.1377å£ããæ2023幎8æ19æ¥ 09:35
å£ããææ§ãããã«ã¡ã¯ã
s=561ã§ããïŒ
N(N-1)(N-2)(N-3)ã»ã»ã»(N-560)=(N^561-N)-a1(N^560-N)+a2(N^559-N)ã»ã»ã»ã»a561(N^2-N)
ãšãŠããããªèšç®ã¯ç¡çã§ãã
No.1378ããããã¯ã¡ã¹ã2023幎8æ19æ¥ 12:08
ãããç§ã以åã¯ãããããŠããŸããããDD++ããã¯åãã§ãããã
https://www.wolframalpha.com/input?i=Table%5B%28N%5E561-N%29mod561%2C%7BN%2C2%2C30%7D%5D&lang=ja
pythonãšããšéã£ãŠæ¡éãã®èšç®åãªãã§ãããå ã¿ã«ãä»åã®ã«äœ¿ãããã©ããã¯å
šãèããŠããŸããã
No.1379å£ããæ2023幎8æ19æ¥ 12:19
DD++æ§ã®èšç®ã¯ã
N=2ã30ã«ãããŠ(N^561ãŒN) mod 561ãæ±ãããã®ã§ãã
No.1380ããããã¯ã¡ã¹ã2023幎8æ19æ¥ 14:35
ããããã¯ã¡ã¹ããããããã«ã¡ã¯ã
ããèŠããšã以åãšã¯éãæ³åã§ããã
ïŒã€ãŸãããªãçå±ãééãããšãããšãå³èŸºã¯ãsãåææ°ã®å Žå(N^t-N){ãã ããtã¯s以äžã®ãã¹ãŠã®èªç¶æ°}ã®ä¿æ°ãsã®åæ°ã«ãªããªããšããã®ããååã®çµè«ã§ããã
ãã ãã(N^t-N)ãtã®åæ°ãšããããæãç«ã€ããã§ãã
(%i1) factor(4-6*3+11*2);
(%o1) 2^3=4X2
(%i2) factor(6-15*5+85*4-225*3+274*2);
%o2) 2^4 3^2=6^2x2^2
(%i3) factor(8-28*7+322*6-1960*5+6769*4-13132*3+13068*2);
(%o3) 2^7 3^3 5=8x2^4x3^3x5
s=4ã®ãšãã
N(N-1)(N-2)(N-3)=(N^4-N)-6(N^3-N)+11(N^2-N)-----(2)
=N(N^3-1)-6N(N^2-1)+11N(N-1)
s=6ã®ãšãã
N(N-1)(N-2)(N-3)(N-4)(N-5)=(N^6-N)-15(N^5-N)+85(N^4-N)-225(N^3-N)+274(N^2-N)------(3)
ã§ããããã(1)ããã
=N(N^5-1)-15N(N^4-1)+85N(N^3-1)-225N(N^2-1)+274N(N-1)
s=8ã®ãšãã
N(N-1)(N-2)(N-3)(N-4)(N-5)(N-6)(N-7)=(N^8-N)-28(N^7-N)+322(N^6-N)-1960(N^5-N)+6769(N^4-N)-13132(N^3-N)+13068(N^2-N)------(4)
ã§ããããã(1)ããã
=N{(N^7-1)-28(N^6-1)+322(N^5-1)-1960(N^4-1)+6769(N^3-1)-13132(N^2-1)+13068(N-1)}
以åãšéã£ãŠãä¿æ°ã«ææ°ããããŠç·åãåã£ãŠããã®ã§ããã
No.1381å£ããæ2023幎8æ19æ¥ 14:54
9é建ãŠã®ãã«ããããããã«ã©ããã®3ãµæã®éã«ããæ¢ãŸããªã12å°ã®ãšã¬ããŒã¿ãèšçœ®ãã
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ãšã¬ããŒã¿E1,E2,,E12ãæ¢ãŸãéã3ã€ããããæå®ããŠã¿ãŠãã ããã
No.1371GAI2023幎8æ15æ¥ 18:26
ãã®ãšã¬ããŒã¿ã®åé¡ãããããã«å³ããæ¡ä»¶ã®åé¡ã
ãç§ã®åå¿é² > æ¡åŒµã«ãŒã¯ãã³åé¡ã
ã®äžã«ãã¬ã«ãŒã¯ãã³åé¡ãšããŠèŒã£ãŠããŸãã
ãã¬ã«ãŒã¯ãã³åé¡ã®è§£ã¯ãããæ¡ä»¶ã®ç·©ããã®ãšã¬ããŒã¿åé¡ã®è§£ãšããŠãé©ããŸãã
No.1372ããã²ã2023幎8æ16æ¥ 01:19
ãšã¬ããŒã¿åé¡ãšã«ãŒã¯ãã³ã®å¥³çåŸåé¡ã¯é£åããŠãããã§ããã
äžè¬ã«ãšã¬ããŒã¿ã®ç·æ°ã(2*n+1)*(3*n+1)ã§åãšã¬ããŒã¿ã¯ã©ããã®
éã®3ãæã§çšŒåããããã«åããšããã©ã®éãããä»ã®éã«è¡ããããã«
ãªãããã®ãã«ã®é«ãã®æ倧éæ°ã¯6*n+3ãšãªãã
ãããf((2*n+1)*(3*n+1),3)=6*n+3ãã§è¡šããŠããã
ããããn=1,2,3,4ã§åœãŠã¯ãããš
n=1ã§12å°ã®ãšã¬ããŒã¿ã§ã¯9FãŸã§ã®ãã«(åºé¡ã®åé¡)
n=2ã§35å°ã®ãšã¬ããŒã¿ã§ã¯15FãŸã§ã®ãã«(ã«ãŒã¯ãã³ã®å¥³çåŸã®è§£ãå©çšã§ããã)
n=3ã§70å°ã®ãšã¬ããŒã¿ã§ã¯21FãŸã§ã®ãã«
n=4ã§117å°ã®ãšã¬ããŒã¿ã§ã¯27FãŸã§ã®ãã«
ã«å¯ŸããŠèšèšã§ããã
ãŸãä»ã«ã
f(s^2+s,s)=s^2ã®ãããªãã®ã,
ãšã¬ããŒã¿ç·æ°ãs^2+sã§ãã«ã®é«ããs^2éãŸã§ã®æã¯
åãšã¬ããŒã¿ãã©ããsåã®éã ãããæ¢ãŸããªãåããããã°
ã©ããªéããã§ãä»ã®éãžè¡ãããšã¬ããŒã¿ãéè¡ã§ããã
ããããs=2,3,4,5,6,7ãã
f(6,2)=4
f(12,3)=9
f(20,4)=16
f(30,5)=25
f(42,6)=36
f(56,7)=49
以åç§ã®åå¿é² > æ¡åŒµã«ãŒã¯ãã³åé¡ã§
ããã²ããããæçš¿ãããŠãã
æ¹é£[â»]
0 1 2 3 4 5 6
7 8 9 10 11 12 13
14 15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31 32 33 34
35 36 37 38 39 40 41
42 43 44 45 46 47 48
éæ¹é£[1]
3 34 9 40 15 46 21
28 10 41 16 47 22 4
11 35 17 48 23 5 29
36 18 42 24 6 30 12
19 43 25 0 31 13 37
44 26 1 32 7 38 20
27 2 33 8 39 14 45
éæ¹é£[2]
38 6 16 33 43 11 21
0 17 34 44 12 22 39
18 28 45 13 23 40 1
29 46 7 24 41 2 19
47 8 25 35 3 20 30
9 26 36 4 14 31 48
27 37 5 15 32 42 10
éæ¹é£[3]
17 13 2 47 36 32 21
7 3 48 37 33 22 18
4 42 38 34 23 19 8
43 39 28 24 20 9 5
40 29 25 14 10 6 44
30 26 15 11 0 45 41
27 16 12 1 46 35 31
ã®åæ¹é£ã®åè¡ãååãå©çšãããŠè²°ããšãã®f(56,7)=49
ã®ã¢ãã«ãäœãããšãåºæ¥ãŸããã
远䌞ïŒ
ãã®ã¢ãã«ãã€ãã£ãŠããã°
f(s^2-s+1,s)=s^2-s+1
ã§ã®s=8
å³ã¡f(57,8)=57ã§ã®ã¢ãã«
å
šéšã§57å°ã®ãšã¬ããŒã¿ã57é建ãŠã®ãã«ã«èšçœ®ã
åãšã¬ããŒã¿ãã©ããã®éã®8ãµæã皌åããããã«äžæãçµã¿åãããŠããã°
ã©ã®éãããä»»æã®éãžéè¡ããŠãããšã¬ããŒã¿ãååšããŠããèšèšãé£ãªãåºæ¥ãã
(äžã®ã¢ãã«ã®è¡ãšåãå
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ãããã£ãŠããã²ãããã®s:çŽ æ°ã§ã®
f(s^2+s,s)=s^2ã®ã¢ãã«ãæ§æã§ããæ±çšçæ§ææ¹æ³ã䜿ãã°
f((s+1)^2-(s+1)+1,s+1)=(s+1)^2-(s+1)+1
å³ã¡
f(13,4)=13
f(31,6)=31
f(57,8)=57
f(133,12)=133
f(183,14)=183
f(307,18)=307
f(381,20)=381
f(553,24)=553
f(871,30)=871
f(993,32)=993
f(1407,38)=1407
f(1723,42)=1723
f(1893,44)=1893
f(2257,48)=2257
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