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(1349^3+675^3)/(1349^3+674^3) =
(1349+675)/(1349+674) =
2024/2023
ãããªããšãã§ããã®ã¯ã©ããªå¹Žã§ãããïŒ
æ¥å¹Žãªã
(4049^3+2026^3)/(4049^3+2023^3) = (4049+2026)/(4049+2023) = 2025/2024
ã¿ãããªããšã§ãããïŒ
ä»»æã®å¹Žã§ã§ãããããªïŒ
nâZ, nâ¡1(mod 3)ã®ãšã
ãã[a,b,c]=[(2*n+1)/3,(n+2)/3,(n-1)/3]
ã¯ã
ã a,b,câZ,
ã (a^3+b^3)/(a^3+c^3)=(a+b)/(a+c)=(n+1)/n,
(a+b)=(a+c)+1
ãæºããã
ãŸããnâZ-{0}ã«å¯ŸããŠã
ã (a^3+b^3)/(a^3+c^3)=(a+b)/(a+c)=(n+1)/n,
(a+b)=(a+c)+1
ãæŽæ°è§£[a,b,c]ãæã€ãªãã°ãnâ¡1 (mod 3)ã§ããã
以äžãããæçµè¡ã®åæ¯nããnâ¡1 (mod 3)ããæºãããšãã§ããããã®ãšãã«éãå¯èœã
æçåŒ
((x+y)^3+y^3)/(x^3+(x+y)^3) = ((x+y)+y)/(x+(x+y))
ããåºçºããŸããã
x+2y=2024
2x+y=2023
ãèŠè«ããŠ
x=675
y=674
ãåŸãŸããã
ãã®ããæ¹ã§ã¯
3x+3y=2023+2024
ã§å³èŸºã3ã®åæ°ãšãªãããŸããããŸãã
H.Nakao ããããã¯ãã¡ãã®ã«ãŒããå³å¯ã«ç€ºããŠããã ããŸããã
äžæ¹ã«ãã㊠DD++ ããã«ãããæ瀺ã«ã¯
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çŽåâŠâŠãããŸãåããŠããŸãã
詳ãã解説ããé¡ãç³ãäžããŸãã
ããããDengan ããã«ä¹ã£ãã圢ã§ã
2 çªç®ã®èŸºãèŠçŽåæ°ã§ããå¿
èŠã¯ãªãã®ã§ã
x+2y=2024k
2x+y=2023k
ã§ããã°ããã
3x+3y=(2023+2024)k
ãšãªã£ããšãã«æ¬åŒ§å
ã 3 ã®åæ°ã§ãªããŠã k ã 3 ã®åæ°ã§ããã°äœãåé¡ãªããšããã ãã®è©±ã§ãã
ãšããã§ã
(7^3+7^3)/(8^3+5^3) = (7+7)/(8+5) = 14/13
ã¿ãããªãã¿ãŒã³ã£ãŠãã®åé¡ã«ãããŠã¢ãªã§ãããïŒ
4 æ°ãã©ãã©ãããã ãšããåæã§ã
(3N+2)/(3N+1) å以å€ãäœãã¯ãããã®ã®ããã®åœ¢ã®æ¹ãããããªåŒã§äœããããšãå€ãããã§ããã
(1025^3+999^3)/(1034^3+989^3) = (1025+999)/(1034+989) = 2024/2023
(1025^3+999^3)/(1034^3+989^3) = (1025+999)/(1034+989) = 2024/2023
倧å€ã«èå³æ·±ãã§ãã
a*b = c*d ãèŠãã 4 æ° a, b, c, d ã«å¯Ÿãã
p = a+b-c, q = a+b-d, r = c+d-a, s = c+d-b ãšãããšã
(p^3+q^3)/(r^3+s^3) = (p+q)/(r+s) ãæãç«ã¡ãŸãã
â» p, q, r, s ãæŽæ°ã§ããã°ãa, b, c, d ãæŽæ°ã§ããå¿
èŠã¯ãããŸããã
â» a = (2x+y)/3, b = 2(x+2y)/3, c = (x+2y)/3, d = 2(2x+y)/3 ãšããã°ãDengan ãããçšããåŒã«ãªããŸãã
(a+b)-(c+d) = k ã§ãããšãã(p+q)-(r+s) = 3k ã«ãªãã®ã§ã
k = 1/3 ã«ãªãããã«ããããp+q ã 3 ã®åæ°ã«ãªãããã«ããªãã k = 1 ã«ãããã§ã
ä»åã®ç®çã®ããã«ååãåæ¯ãã 1 ã ã倧ããåæ°ãåŸãããŸãã
ãŸãã(a+b)-(c+d) = k ãèŠè«ããå Žåã
p = a+b-c = d+k, q = a+b-d = c+k, r = c+d-a = b-k, s = c+d-b = a-k ãšãªããæŽæ°ã«ãã調æŽãå€å°æ¥œã«ãªããŸãã
äŸ1
(a+b)-(c+d) = 1/3 ãš a*b = c*d = 400/9 ãèŠè«ããŠã
a = 16/3, b = 25/3, c = 20/3, d = 20/3 ãšãããšã
p = 7, q = 7, r = 8, s = 5 ãåŸããã14/13 ãäœããŸãã
äŸ2
(a+b)-(c+d) = 1/3 ãš a*b = c*d = 1120/9 ãèŠè«ããŠã
a = 28/3, b = 40/3, c = 32/3, d = 35/3 ãšãããšã
p = 12, q = 11, r = 13, s = 9 ãåŸããã23/22 ãäœããŸãã
äŸ3
(a+b)-(c+d) = 1 ãš a*b = c*d = 2*2024*2025 ãèŠè«ããŠã
a = 2024, b = 4050, c = 2025, d = 4048 ãšãããšã
p = 4049, q = 2026, r = 4049, s = 2023 ãåŸããã6075/6072 = 2025/2024 ãåŸãããŸãã
äŸ4
(a+b)-(c+d) = 1/3 ãš a*b = c*d = 2*2023*2024/9 ãèŠè«ããŠã
a = 2023/3, b = 4048/3, c = 2024/3, d = 4046/3 ãšãããšã
p = 1349, q = 675, r = 1349, s = 674 ãåŸããã2024/2023 ãåŸãããŸãã
äŸ5
(a+b)-(c+d) = 1/3 ãš a*b = c*d = 28*29*106*107/9 ãèŠè«ããŠã
a = 2968/3, b = 3103/3, c = 2996/3, d = 3074/3 ãšãããšã
p = 1025, q = 999, r = 1034, s = 989 ãåŸããã2024/2023 ãåŸãããŸãã
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