1ãã7ãŸã§ã®æ°åãäžåºŠã ã䜿ãããšãæ¡ä»¶ã«
(1)4æ¡ã®æŽæ°abcdãš3æ¡ã®æŽæ°efgã足ããšçŽ æ°ãã§ãããšãã
çµåã(a,b,c,d,e,f,g)ã¯äœéãã§ãã®åºæ¥ãçŽ æ°ã®çš®é¡ã¯äœéããããïŒ
(2)aÃbÃcÃd + eÃfÃg ãçŽ æ°ãã§ãããšã
ãã®çµåã(a,b,c,d,e,f,g)ã¯äœéãã§ãã®åºæ¥ãçŽ æ°ã®çš®é¡ã¯äœéããããïŒ
ã©ãªãã
æ°åã®1ãã18ãŸã§ãäžã€ãã€äœ¿ã£ãŠ
9åã®1ããå°ããæ¢çŽåæ°ãäœãããšã
åºæ¥ãçµåããäœéãåºæ¥ãã調ã¹ãŠãããŸãããïŒ
èšç®ã§æ±ãŸãã®ãïŒ
ã³ã³ãã¥ãŒã¿ã§ããã°ã©ã çã«åŠçã§ããã®ãïŒ
ã©ã¡ãã§ãæ§ããŸããã®ã§ãé¡ãããŸãã
https://oeis.org/A009679
âããã§æ£ãããã°
59616éãã ãšæããŸãã
èšç®ã§åºãã®ã¯é£ãããšæããŸãã
# n=2,4,6,âŠ,18ã«å¯Ÿããçµåãæ°ãããã°ã©ã ã§èª¿ã¹ãçµæã®æ°åãæ€çŽ¢ãããäžèŽãããã®ããããŸããã
äžçäžã§ã¯èª°ãªã£ãšèª¿ã¹äžããŠãããã®ã§ããã
ç§ã¯äžé±éãããè©Šè¡é¯èª€ãç¹°ãè¿ããããã°ã©ã ãè²ã
æžãçŽããã£ãšå
šéšã§58320éããæ±ãããšæã£ãŠãããã§ãã
ã©ããã«èŠèœãšããããã®ããªïœã
èªåã§ãäœåºŠãèŠçŽãããã§ãããããåãããªãã®ã§ãã
ïŒèãæ¹ïŒ
å
šéšãèŠçŽãªåæ°ã§ãããããååãšåæ¯ã¯å¥æ°ãå¶æ°ã§ã®çµåãããšãªãã®ã§
A=[2,4,6,8,10,12,14,16,18]
B=[1,3,5,7,9,11,13,15,17]
ã®2çµã«åã
AãšBãçµåãããŠã§ãã1ããå°ããæ¢çŽåæ°ã¯æ¬¡ã®69åã§
M=[1/2, 2/3, 2/5, 2/7, 2/9, 2/11, 2/13, 2/15, 2/17,
1/4, 3/4, 4/5, 4/7, 4/9, 4/11, 4/13, 4/15, 4/17,
1/6, 5/6, 6/7, 6/11, 6/13, 6/17,
1/8, 3/8, 5/8, 7/8, 8/9, 8/11, 8/13, 8/15, 8/17,
1/10, 3/10, 7/10, 9/10, 10/11, 10/13, 10/17,
1/12, 5/12, 7/12, 11/12, 12/13, 12/17,
1/14, 3/14, 5/14, 9/14, 11/14, 13/14, 14/15, 14/17,
1/16, 3/16, 5/16, 7/16, 9/16, 11/16, 13/16, 15/16, 16/17,
1/18, 5/18, 7/18, 11/18, 13/18, 17/18]
ããããçµåãããŠèª¿ã¹ãã®ã¯å¯Ÿè±¡ãå€ãããã®ã§æåãããŠããã
(1)1/2ã䜿ããšããã°æ®ãã¯1,2ã®æ°åããµããŸãªãç©ããæ§æããã®ã§
Q1=[ 3/4, 4/5, 4/7, 4/9, 4/11, 4/13, 4/15, 4/17]
Q2=[5/6, 6/7, 6/11, 6/13, 6/17]
Q3=[ 3/8, 5/8, 7/8, 8/9, 8/11, 8/13, 8/15, 8/17]
Q4=[3/10, 7/10, 9/10, 10/11, 10/13, 10/17]
Q5=[5/12, 7/12, 11/12, 12/13, 12/17]
Q6=[ 3/14, 5/14, 9/14, 11/14, 13/14, 14/15, 14/17]
Q7=[3/16, 5/16, 7/16, 9/16, 11/16, 13/16, 15/16, 16/17]
Q8=[5/18, 7/18, 11/18, 13/18, 17/18]
ã«åããŠçœ®ã
for(n1=1,8,Q1[n1])
for(n2=1,5,Q2[n2])
for(n3=1,8,Q3[n3])
for(n4=1,6,Q4[n4])
for(n5=1,5,Q5[n5])
for(n6=1,7,Q6[n6])
for(n7=1,8,Q7[n7])
for(n8=1,5,Q8[n8])
ã§ããããã®ã°ã«ãŒãããæ¢çŽåæ°ãçµåãããŠããã
ãã®çµåãããç°ãªã16åã®ç°ãªãæ°åããäœãããŠããã°ã«ãŠã³ãããŠéèšããã
ããããŠæ±ãŸã£ãéèšãs1=4788ãåŸãããã
åæ§ãªèãæ¹ã§
(2)2/3ã䜿ããšããã°æ®ãã¯2,3ãå«ãŸãªãç©ããçµã¿åãããŠããã®ã§
Q1~Q8ãããã«åãããŠéžã³çŽããæ¡ä»¶ãæºãããã®ãã«ãŠã³ãããŠãããš
s2=9144
以äžåæ§ã§
(3)2/5ã䜿ãå Žå
s3=5328
(4)2/7ã䜿ãå Žå
s4=5184
(5)2/9ã䜿ãå Žå
s5=9144
(6)2/11ã䜿ãå Žå
s6=4788
(7)2/13ã䜿ãå Žå
s7=4788
(8)2/15ã䜿ãå Žå
s8=10368
(9)2/17ã䜿ãå Žå
s9=4788
以äžããæ±ããç·æ°ã¯
4788*4+5184+5328+9144*2+10368=58320éã
ãªã®ã§ãã
ãããæ£è§£ãšæããã59616éããªã®ã§1296éãã¯äœåŠãžè¡ã£ãã®ã?
(8)2/15ã䜿ãå Žå
ã®å€ã¯10368ã§ãªã11664ã§ãã
ãã®ä»ã¯ãã¹ãŠåã£ãŠããŸããã
2/15ã®ãã§ãã¯ãããã
Q4=[[1, 10], [3, 10], [7, 10], [9, 10], [10, 11], [10, 13], [10, 17]]
ã«å¯Ÿãã
for(n4=1,7,Q4[n4])ã§ã®ãã§ãã¯ã§ããã¹ãæã
for(n4=1,6,Q4[n4])
ãšééã£ãŠåŠçããŠããããšãåå ã§ããããšãå€æããŸããã
ééããªãæ§ã«æ
éã«é²ãã§ãã調æ»ãçµç€ã«è¿ããªã£ãé ã«ã¿ã€ããã¹
ãçºçãããŠããŸã£ãŠããã
å床for(n4=1,7,Q4[n4])ã§ã®ç¢ºèªã§ã¯s7=11664ã§äžèŽåºæ¥ãŸããã
ééããã®ã«æ°ä»ãããã«ã¯ãééã£ãããšã確èªããåŸã§ããèŠããŠããªãã®ã§åä»ã§ãã
Q1,Q2,âŠãããã°ã©ã ã§çæãããã®åæ°ãïŒããã°ã©ã ã§èªåçã«ïŒã«ãŒãã«ãŠã³ãã«ããã°ééããããšã¯ãªããªããšæããŸãã
人ãããã°ã©ã ãäœã£ãŠããã®ãåèã«ããŠããã
ãããªã«äœæ¥ãèããŠãã£ãšæã«å
¥ããæ°å€ã
gp > matpermanent(matrix(9,9,i,j,gcd(2*i,2*j-1)==1))
%192 = 59616éã
ã®äžçºã§è¡ãããšããã
åŸã£ãŠ1ïœ32ã®æ°åãããããäžåºŠãã䜿çšã§ããªã
16åã®1ããå°ããæ¢çŽåæ°ã(4Ã4è¡åãäœãã)
äœããçµåãã®ç·æ°ã¯
gp > matpermanent(matrix(16,16,i,j,gcd(2*i,2*j-1)==1))
%193 = 768372168960éã
ãããšãŠããŸãšãã«ã¯æ±ããããšãé£ãã倧ããã§ããããšãç°¡åã«ããããšããïŒïŒ
ã»ããšã«æ°åŠã®åïŒäžã«ããæ°åŠçæ§é ã®äžæè°ã)ãæããŸãã
èšç®ã§äžè¬çã«åºãã®ã¯é£ãããšããŠããä»åã®æ°å€èšå®ã«éããªããã»ã©é£ãããªãã§ããã
ãšãããã 2 ãš 3 ã§çŽåã§ããªãããã«åæ°ãäœãæ¹æ³ã¯ã
6 ã®åæ°ã䜿ã 3 çµã3 ã®åæ°ãš 2 ã®åæ°ã䜿ã 3 çµãæ®ã 3 çµã®é ã«èããŠã
6P3 * 6P3 * 3! = 86400 éã
ãã®äžã§ã
5/10 ãã§ããŠããŸãçµã
5P3 * 5P3 * 2! = 7200 éã
10/15 ãã§ããŠããŸãçµã
6P3 * 5P2 * 3! = 14400 éã
7/14 ãã§ããŠããŸãçµã
5P3 * 5P3 * 2! = 7200 éã
5/10 ãš 7/14 ããšãã«ã§ããŠããŸãçµã
4P3 * 4P3 * 1! = 576 éã
10/15 ãš 7/14 ããšãã«ã§ããŠããŸãçµã
5P3 * 4P2 * 2! = 1440 éã
ãã£ãŠãããããæ¢çŽåæ°ã«ãªãçµã¯
86400-7200-14400-7200+576+1440 = 59616 éã
ãã¿ãŸããDD++ãã
説ææãšããã瀺ãåŒãç§ã®é ã§ã¯äœãã©ãèãããæãç«ã€ã®ãå
šãèŠããŠããŸããã
äŸãã°
6 ã®åæ°ã䜿ã 3 çµã3 ã®åæ°ãš 2 ã®åæ°ã䜿ã 3 çµãæ®ã 3 çµã®é ã«èããŠã
6P3 * 6P3 * 3! = 86400 éã
ã¯ãŠïŒ
6P3ã®6ã¯äœãæãã6ãªã®ãïŒ
3!ã¯äœããã£ãŠ3!ã«ããã®ãïŒ
以äžåŸã®æ¹ã®èª¬ææãšåŒã®æå³ãå
šãèªã¿åããªãããŸãã
èšç®ãšããããå°ãããçµæã¯æ£ããã®ã ãããããããã«ã¯å°ãçç±ãæã€
æ ¹æ ãããäºã¯ç解ã§ãããã§ãããå
šäœãéããæ§æ³ã®ãããããèŠããŠããªã
äºãªãã ãšæããŸãã
ããå°ã解説ãå
¥ããŠèª¬æé¡ããŸãããïŒ
18 ãŸã§ã®äžã«ã¯å¶æ°ã 9 åããã®ã§ã2 ã§çŽåã§ããªãããã«ã¯å
šãŠå¶æ°ãšå¥æ°ã§å¯Ÿã«ããŠåæ°ãäœãå¿
èŠããããŸãã
ãããŠã3 ã§ãçŽåã§ããªãããã«å¯Ÿã«ããã«ã¯ã
ã»6 ã®åæ°ïŒ3 ã€ããïŒã«ã¯ 3 ã§å²ãåããªãå¥æ°ïŒ6 ã€ããïŒã®äžãã 3 ã€éžãã§å²ãåœãŠã
ã»3 ã®åæ°ã§ããå¥æ°ïŒ3 ã€ããïŒã«ã¯ã3 ã§å²ãåããªãå¶æ°ïŒ6 ã€ããïŒã®äžãã 3 ã€éžãã§å²ãåœãŠã
ã»æ®ã£ã 3 ã€ãã€ã¯ãé©åœã«å¶æ°ãšå¥æ°ã®ãã¢ã 3 çµäœã
ããšã«ãªããŸãã
ã ãã 6P3 * 6P3 * 3! = 86400 éããšãªããŸãã
ãã以åŸããæå®ã®åæ°ãæåã«äœã£ãŠããŸãããšã«ãããšäœåããäœåéžã¶ããå€ãããŸãããèãæ¹ã¯åãã§ãã
ããã³ã®å
¬åŒãšããŠ
Ï/4=4*atan(1/5)-atan(1/239)
ãæåã§ããã
äžè¬ã«tanã®4åè§ã®å
Œ΋
tan(Ξ)=Tã®æ
tan(4*Ξ)=(4*T-4*T^3)/(1-6*T^2+T^4)
ãªã®ã§
ä»tan(Ξ)=1/5 <=> Ξ=atan(1/5)
ãšããã°
tan(4*Ξ)=(4/5-4/5^2)/(1-6/5^2+1/5^4)=120/119
ãããã
4*Ξ=atan(120/119)
å³ã¡
4*atan(1/5)=atan(120/119)
ãŸãäžè¬ã«
-atan(Ξ)=atan(-Ξ) ãã
ããã³ã®å
Œ΋
atan(120/119)+atan(-1/239)=Ï/4
ãšè¡šèšããããšãå¯èœ
æŽã«Degan ããããæ瀺ããã
atan(a/b)+atan(s/t)=atan((a*y+b*x)/(b*y-a*x))+atan((s*y-t*x)/(t*y+s*x))
ãæ¡ä»¶ãç¡èŠããŠ
H(a,b,s,t,x,y)=[(a*y+b*x)/(b*y-a*x),(s*y-t*x)/(t*y+s*x)]
ã§ã®èšç®ãx,yãåæã«éžãã§ãã£ãŠã¿ãã
gp > atan(120/119)+atan(-1/239)
%2817 = 0.78539816339744830961566084581987572105
gp > Pi/4
%2818 = 0.78539816339744830961566084581987572105
ããã§èªç±ã«x,yãéžãã§å€æããŠãããš
gp > H(120,119,-1,239,1,2)
%2819 = [359/118, -241/477]
gp > atan(359/118)+atan(-241/477)
%2823 = 0.78539816339744830961566084581987572105
gp > H(120,119,-1,239,1,3)
%2820 = [479/237, -121/358]
gp > atan(479/237)+atan(-121/358)
%2824 = 0.78539816339744830961566084581987572105
gp > H(120,119,-1,239,1,4)
%2821 = [599/356, -243/955]
gp > atan(599/356)+atan(-243/955)
%2825 = 0.78539816339744830961566084581987572105
gp > H(120,119,-1,239,11,17)
%2822 = [3349/703, -1323/2026]
gp > atan(3349/703)+atan(-1323/2026)
%2826 = 0.78539816339744830961566084581987572105
x>yãšããŠã
gp > H(120,119,-1,239,7,3)
%2828 = [-1193/483, -838/355]
gp > atan(-1193/483)+atan(-838/355)
%2829 = -2.3561944901923449288469825374596271632
ãã®æã¯Piãè£ã
gp > Pi+%2829ãšããŠããã°
%2831 = 0.78539816339744830961566084581987572106
gp > H(120,119,-1,239,1/7,1/3)
%2833 = [1197/473, -362/835]
gp > atan(1197/473)+atan(-362/835)
%2834 = 0.78539816339744830961566084581987572105
ã§ã©ãã§ãÏ/4ãæ§æãããŸããã
(远䌞)
16次ã®arctanç³»éæ¹é£ã¯çµå±arctanã§ã®å€ã¯éæ¹é£ãã§ãããããããäžããåæ¢çŽåæ°ã¯
åãæ°åãå«ãã§ããŸããã®ããäœããŸããã§ãããïŒã©ãªãã解æããŠäžããã)
åããã®ãã¿ãããŸããã®ã§ã玹ä»ããããŸãã
次ã®ããŒãžã«ã¯ã256åã®æçæ°ã
16ïœ16 ã®æ¹é£ãšããŠæäŸãããŠããŸãã
https://github.com/TokusiN/AtanMagic/blob/main/data.txt
ãã®256åã®æçæ°ã§ã¯ã1ãã512ãŸã§ã®èªç¶æ°ãå
šãŠäœ¿ãããã®ãã¡256åã®èªç¶æ°ãååã«ãæ®ãã®256åãåæ¯ãšããŠäžå¯Ÿäžã«çµã¿åãããã§ãã256åã®åæ°ããã®ãŸãŸæ¢çŽã«ãªã£ãŠããŸããèšãæããã°çŽåæäœã¯äžåãããŠããŸããã
åè¿°ã®éãã«ã16ïœ16 ã®æ¹é£ãšããŠããŒã¿æäŸãããŠå±
ãããã§ããã
åãã¹ç®ã®æçæ°ã«arctané¢æ°ãäœçšãããå€ã§æ¹é£ãã€ãããšããããéæ¹é£ã«ãªã£ãŠããŠã瞊暪æãããããã®ç·åã 2Ï ã«ãªã£ãŠããŸãã
éæ¹é£ã§ããããšãåç»åãããã®ã以äžã«ãããŸãã
https://tatt61880.github.io/AtanMagic/
ããã«ããŠãã©ãããçºæ³ãªãã§ãããããâŠâŠ
æèŠããŸããã
äœè
ãã³ãã«ããŒã ïŒTakusiNããç©åãã§ããã
1ïœ512ã®ãã¹ãŠã䜿ã£ãŠ16*16=216åã®æ¢çŽåæ°ãäœãã®ãããããã©ããããarctanã§éæ¹é£ã ãªããŠãã£ããŸãïœã§ãã
ãªããšããã®äººã«é£çµ¡ããšããŠçºèŠã®çµç·¯ãèããããã®ã§ããã
ã³ã³ãã¥ãŒã¿ã§ã®è
åã§ã®è§£æ±ºã趣å³ã ãšæžãããŠã¯ãããäœãããã®æ³åãæ°ä»ããç¡ããã°å°åºç¡çã§ãããã
ãããŒäžã®äžåã人ãããããã ã
ã¡ãªã¿ã«ç¬¬äžè¡ã®åæ°ãéåããŠã¿ãŠããæ°ãè«å€§éããŠäœã«ããã³ããåããŸããã§ããã
èŠçŽåæ°ãšarctanå€ãããããã©ã³ã¹ããšããŠããããšã«æåããã®ã§
ã§ã¯1ïœ8ã®æ°åãäžåãã€äœ¿ã£ãŠã2è¡2åã®1ããå°ãã4ã€ã®æ¢çŽåæ°ã§
å¯èœãªéãçµåããäœã£ãŠã¿ãã
[1/2 3/4]ã
[5/6 7/8]
----------
[1/2 3/4]
[5/8 6/7]
----------

-----------
[7/8 5/6]
[3/4 1/2]
çã®å
šéšã§432éãã®è¡åãäœããŸããã
ãšããããã¹ãŠã®è¡åã§ããã®å€ã«å¯Ÿããarctanã®å€ãåã£ãè¡åã«ã¯äœã®çŸããèŠåã瀺ããã®ã¯ãããŸããã§ããã
ãã£ãŠã¯ããŸããããã¶ã3次æ£æ¹è¡åã§1ïœ18ãçšãã9åã®æ¢çŽåæ°ã§ã®ã¿ã€ãã§ãäœã®ææãããåŸãããªããã®ãšæãããŸãã
16次ãŸã§æ¡åŒµããè¡åã§ãã®åãæ³åãçŸãåºãããšãçã
èå³ãåŒãç«ãŠãŸãã
ïŒSNS ã®X(æ§twitterïŒã§äœè
ã®TokusiNããã«ã©ããã£ãŠäœã£ãã®ãå°ããŠããã
æãä»ããã®ã¯ããã³ç³»ã®å
¬åŒããã§å
·äœçãªäœææ¹æ³ã¯ã¬ãã¬ãã®ã³ã³ãã¥ãŒãã£ã³ã°ãšã®ããšã§ããã
èããšããã«ããã°
ä»»æã®æ£ã®æŽæ° q, r ã«ã€ããŠãµãã€ã®å
¬åŒ
â : Ï/4 = arctan(q/(q +r)) +arctan(r/(2*q +r))
â¡: Ï/4 = arctan(q/(q -r)) +arctan(r/(2*q -r))
ããšãã«æãç«ã€ã®ã ããã§ã
arctané¢æ°ãžã®åŒæ°ã¯ãšãã«çåæ°ã§ãããšãããããœã§ããããã®atanmagic ã§ããããªã£ãŠããŸãã®ã§äœ¿ããããïŒ
Q[1]ããQ[8]ãŸã§ãR[1]ããR[8]ãŸã§ã®16åã®æŽæ°ãããããŒã«éžã¶ãš
8*(Ï/4) =
arctan(Q[1]/(Q[1] +R[1])) +arctan(R[1]/(2*Q[1] +R[1]))
+arctan(Q[2]/(Q[2] +R[2])) +arctan(R[2]/(2*Q[2] +R[2]))
+arctan(Q[3]/(Q[3] +R[3])) +arctan(R[3]/(2*Q[3] +R[3]))
+arctan(Q[4]/(Q[4] +R[4])) +arctan(R[4]/(2*Q[4] +R[4]))
+arctan(Q[5]/(Q[5] +R[5])) +arctan(R[5]/(2*Q[5] +R[5]))
+arctan(Q[6]/(Q[6] +R[6])) +arctan(R[6]/(2*Q[6] +R[6]))
+arctan(Q[7]/(Q[7] +R[7])) +arctan(R[7]/(2*Q[7] +R[7]))
+arctan(Q[8]/(Q[8] +R[8])) +arctan(R[8]/(2*Q[8] +R[8]))
ãšãªãã16åã®arctané¢æ°ã®å€ã®ç·åã 2*Ï
ãšãã圢ãåŸãããŸããatanmagic ã®æ§è³ªã®äžéšã«ãã䌌ãŠããŸããã¡ãã£ãšããã£ãŠããã®ãããããŸããã
äžèšã¯å
¬åŒâ ã®ã¿ã§ã€ã£ã±ã£ãŠããŸãããå®éã«ã¯å
¬åŒâ¡ãšã®æ··åšã§ãããçã§ããèªç±åºŠãèšãäžãããŸããâŠâŠ(é§ç®ïŒè¿œèšãåç
§é¡ããŸã)
ããããäœæŠã§ä»åã®éæ¹é£ãäœãããã®ãã©ãããã ãã§ã¯ãããŸãããâŠâŠæã¿èãããããŸããã
è¿œèšã
â¡ã¯çåæ°ã®ã¿ã®åŒã«ãªã£ãŠããŸããã§ããã謹ãã§ãè©«ã³ããŸãã
â ã¯æ¬¡ã®GAIããã«ãã埡æçš¿å
ã§äœ¿ãããåŒãšå®è³ªçã«åããã®ãšæããŸãã
ç§ãæšæ¥ããã©ããã£ãŠæ§æããŠããããã ããããšããŒãšèãç¶ããŠããŸãã
ç§ãèŠã€ããããã³ã®å
¬åŒãã©ããšããŠãäžè¬ã«1ããå°ããæ¢çŽåæ°ã®s/t
ã«å¯Ÿãããã®ã§(tan(Ï/4)=1ã ããå¯äžæçæ°ãšãªããã®ã§ããã®ãã¿ãŒã³ã¯äœãæãã)
arctan(s/t)+arctan((t-s)/(t+s))=Ï/4ã
ã®çµåãã§å¿
ãÏ/4ãäœããã®ã§,ãããã
arctan(1/2)+arctan(1/3)=Ï/4
arctan(1/4)+arctan(3/5)=Ï/4
arctan(1/5)+arctan(2/3)=Ï/4
arctan(1/6)+arctan(5/7)=Ï/4
arctan(1/7)+arctan(3/4)=Ï/4
arctan(1/8)+arctan(7/9)=Ï/4
arctan(1/9)+arctan(4/5)=Ï/4
arctan(1/10)+arctan(9/11)=Ï/4
åŸã£ãŠããããã¹ãŠè¶³ãåãããã°
A=[1/2,1/3,2/3,1/4,3/4,1/5,3/5,4/5,1/6,1/7,5/7,1/8,1/9,7/9,1/10,9/11]
ã®16åã®æ¢çŽæåã«å¯Ÿãããã®arctanå€ã®åã¯2*Ïãäœãã
gp > vecsum(apply(i->atan(i),A))
%=6.28318530717958647692528(=2*Ï)
ããããµã€ãã«ãã16Ã16次ã®æ£æ¹è¡åã®ç¬¬1è¡ã¯
M1=[5/168 ,259/498 ,216/337 ,129/478 ,381/436 ,266/303 ,6/127 ,78/179
,31/480 ,144/307 ,210/341 ,43/474 ,174/443 ,172/379 ,271/348 ,41/88]
ã§ããã®ã§ããã®æ¢çŽåæ°ãšã»ãããçµãã§Ï/4 ãç£ã¿åºããŠãããã®ã¯
ãããã
5/168 VS 163/173
259/498 VS 239/757
216/337 VS 121/553
129/478 VS 349/607
381/436 VS 55/817
266/303 VS 37/569
6/127 VS 121/133
78/179 VS 101/257
31/480 VS 449/511
144/307 VS 163/451
210/341 VS 131/551
43/474 VS 431/517
174/443 VS 269/617
172/379 VS 207/551
271/348 VS 77/619
41/88 VS 47/129
ã察å¿ããŠããããšãšãªãã
ãšãããå³ã«çŸãã16åã®æ¢çŽåæ°ã¯äœåŠã«ã䜿ãããŠããªããããã163ãªã©ã®æ°ã¯éè€ããŠ
åºçŸããããšãèµ·ãããããã512ããã倧ããªæ°åã䜿ãããããšã«ãªãã
ãã®æ¹éãããã§ã¹ãããããããšã«ãªã£ãã
ããã§ä»è€çŽ æ°ã§ã®åè§ã®æ§åã«åãæ¿ã
å£ã§è§£èª¬ããŠããã®ã倧å€ãªã®ã§ãä»æ€çŽ¢ãæããŠããããã°ã©ã ã§
èªã¿åã£ãŠäžããã
äŸãååšããŠããŠãèšå€§ãªæéãèŠããå¿
èŠããããããªãã§ãã
{t=0;}for(a1=2,512,for(a2=1,a1-1,for(a3=3,512,for(a4=2,a3-1,\
for(a5=4,512,for(a6=3,a5-1,for(a7=5,512,for(a8=4,a7-1,\
for(a9=6,512,for(a10=5,a9-1,for(a11=7,512,for(a12=6,a11-1,\
for(a13=8,512,for(a14=7,a13-1,for(a15=9,512,for(a16=8,a15-1,\
for(a17=10,512,for(a18=7,a17-1,for(a19=11,512,for(a20=10,a19-1,\
for(a21=12,512,for(a22=9,a21-1,for(a23=13,512,for(a24=12,a23-1,\
for(a25=14,512,for(a26=11,a25-1,for(a27=15,512,for(a28=14,a27-1,\
for(a29=16,512,for(a30=13,a29-1,for(a31=17,512,for(a32=16,a31-1,\
if(gcd(a1,a2)==1 && gcd(a3,a4)==1 && gcd(a5,a6)==1 && gcd(a7,a8)==1 && \
gcd(a9,a10)==1 && gcd(a11,a12)==1 && gcd(a13,a14)==1 && gcd(a15,a16)==1 &&\
gcd(a17,a18)==1 && gcd(a19,a20)==1 && gcd(a21,a22)==1 && gcd(a23,a24)==1 && \
gcd(a25,a26)==1 && gcd(a27,a28)==1 && gcd(a29,a30)==1 && gcd(a31,a32)==1 && \
imag((a1+a2*I)*(a3+a4*I)*(a5+a6*I)*(a7+a8*I)*(a9+a10*I)*(a11+a12*I)*(a13+a14*I)*(a15+a16*I)*\
(a17+a18*I)*(a19+a20*I)*(a21+a22*I)*(a23+a24*I)*(a25+a26*I)*(a27+a28*I)*(a29+a30*I)*(a31+a32*I))==0 , \
print(t++";"a2/a1","a4/a3","a6/a5","a8/a7","a10/a9","a12/a11","a14/a13","a16/a15","\
a18/a17","a20/a19","a22/a21","a24/a23","a26/a25","a28/a27","a30/a29","a32/a31)) \
))))))))))))))))))))))))))))))))
äžèšã®ããæ¹ã§ã¯ãšãŠããããªãããããæéããããŠãç¡çãšå€å®
ããã§1ïœ512ãäžåºŠãã€çšããæ¡ä»¶ãé€ãã°ã次ã®ãããª16次ã®è¡åã§ã¯arctanã§ã®åè¡ãååã2ã€ã®å¯Ÿè§ç·ã§ã®åã¯2*Ïã®
éæ¹é£ãšã¯ãªãããã§ãã
éã«èšãã°åŠäœã«1ïœ512ãäžåºŠãã€äœ¿ããšããããšãåãããšãããããŸãã
[163/173 239/757 121/553 349/607 55/817 37/569 121/133 101/257 449/511 163/451 131/551 431/517 269/617 207/551 77/619 47/129]
[299/623 107/531 199/773 191/339 353/467 159/529 93/263 315/457 139/513 5/857 19/331 183/239 357/587 235/263 91/463 323/541]
[133/379 179/627 137/571 301/593 263/431 223/327 151/247 371/641 33/901 157/347 247/255 267/529 33/277 87/773 245/337 151/807]
[269/397 133/869 203/521 297/679 143/599 191/577 173/293 141/449 161/325 197/769 91/367 461/527 107/553 287/723 311/369 173/389]
[ 79/203 361/543 59/763 245/419 279/347 91/313 401/607 359/625 77/207 287/359 47/883 49/739 355/489 199/617 143/353 91/503]
[337/361 23/449 289/373 47/259 409/503 445/561 61/187 31/983 263/457 275/459 249/347 1/829 109/823 191/683 141/367 221/419]
[ 97/141 105/499 131/359 1/557 121/843 201/607 323/503 127/491 199/787 331/583 401/491 261/521 391/577 377/523 37/813 295/383]
[ 31/411 421/557 341/529 249/421 61/383 273/293 237/575 295/503 439/471 31/599 139/885 31/209 163/647 119/729 251/455 377/543]
[ 49/559 213/323 329/431 11/427 113/587 209/609 53/779 199/269 213/757 59/137 311/547 61/329 421/475 239/473 277/607 469/549]
[ 89/817 199/241 175/639 359/519 185/553 203/663 181/395 89/543 73/783 333/557 215/779 139/759 343/607 473/507 275/311 239/701]
[ 97/575 177/703 415/559 237/493 163/317 43/883 353/379 185/303 133/233 49/113 143/745 241/537 113/877 331/469 261/755 255/707]
[157/443 337/413 53/429 99/923 401/507 197/557 125/467 227/553 269/569 349/651 193/805 253/503 113/789 357/589 301/451 289/587]
[ 17/563 223/253 193/199 337/411 213/641 217/419 89/613 467/471 139/653 139/797 199/251 49/457 25/967 167/367 135/181 61/631]
[159/379 283/301 227/727 119/529 61/863 327/341 137/653 201/515 191/331 403/515 187/617 423/581 337/683 25/553 31/815 289/481]
[101/141 53/467 143/691 145/497 175/447 223/517 17/667 127/755 173/189 113/221 191/193 259/281 367/427 101/497 217/557 21/311]
[317/469 49/793 251/373 439/477 283/333 137/415 263/665 29/349 79/639 189/577 351/631 71/203 293/341 145/499 83/387 163/491]
æ£æ° a,b,s,t,x,y ã«ã€ããŠ
a ïŒ b, s ïŒ t, x ïŒ y
ãèŠè«ããŠãããŠ
arctan(a/b) +arctan(s/t)
ïŒ arctan((a*y +b*x)/(b*y -a*x)) +arctan((s*y -t*x)/(t*y +s*x))
ãšã§ããŸãã
x ïŒ 0 ãªãã°å³èŸºãšå·ŠèŸºã¯çãããªãããšã¯èªæã§ãã
巊蟺ã®ïŒåã®æçæ°ãã¡ãã£ãšããããŠã¿ãããšèããã®ã§ããå³èŸºã®arctanã«å°ãããå€ãçåæ°ã«ãªãã«ãããŠèŠæŠããŠãããŸãããŸããããããããšã«çŽåãçºçãããããããŸããã
ãªããªãããŸããããªãã§ãã
æ¯èŒçäœãæã(arctanãšã¿ã€ãããæãçç¥ã§atanã§æžããŠããŸãã)
atan(1/4)+atan(3/5)(=Ï/4) (a=1,b=4,s=3,t=5ã«å¯Ÿå¿)
ããDengan kesaktian Indukmuããã®ããããã¯ããã¯ãçšããŠ
å
ã
ã®16è¡åã®ç¬¬1è¡ã®æåãäœãåºããŠãããš(第2é
ç®ã«çžåœ)
(x,y)=(479,855)->atan(163/173)+atan(5/168)ããã:ok
(x,y)=(38,2041)->atan(349/607)+atan(129/478) :noã
(x,y)=(351,653)->atan(121/133)+atan(6/127) :ok
(x,y)=(133,794)->atan(101/257)+atan(78/179) :ok
(x,y)=(1258,2493)->atan(449/511)+atan(31/480) :ok
(x,y)=(201,1967)->atan(163/451)+atan(144/307) :ok
(x,y)=(71,147)->atan(431/517)+atan(43/474) :no
(x,y)=(27,161)->atan(269/617)+atan(174/443) :no
(x,y)=(277,2411)->(atan(207/551)+atan(172/379) :no
(x,y)=(59,563)->atan(47/129)+atan(41/88) :ok
ãªãåŒã«äœãå€ããããã
ãªã
atan(259/498)
atan(216/337)
atan(381/436)
atan(266/303)
atan(210/341)
atan(271,348)
ã«ã¯(x,y)ãèŠã€ãããªãã
ãã ãåæ¯ã512ãããããã®ã¯æ¡çšããªãããšã«ããã(:no)
第1é
ç®ã®åæ¯ã512以äžãªãæ¡çšããŠ(:ok)ããã§äœ¿ãããŠããéšåã®
å
ã®è¡åãããã®æ°ãå«ãåæ°ã¯æ¶ããŠããã
ãããä»åºŠã¯å
ã®16è¡åã®ç¬¬2è¡ã®åæ°æ¶ãæ®ã£ããã®ã«å¯ŸããŠäœã£ãŠããã
ãããç¹°ãè¿ããŠè¡ãã°èªããš1ïœ512ã ãã§äœãããŠããåæ°ãæ®ã£ãŠãããªãã ãããïŒ
ãªãéæ¹é£ã®äœãæ¹ã«ã€ããŠäœåºŠãå°ããŠã¿ããTokusiNãããã
8次ããé ã«æ€èšããçµæãçŸå®çãªæéã§äœãããšãå¯èœãªæå°æ¬¡æ°ã16ã ã£ãã®ã§ãã
ããã³ç³»ã®ååšçå
¬åŒã®çæææ³ãç解ãããšããã®éæ³é£ã«äœ¿ãããŠããåæ°ãã©ã®ããã«éžã°ãããããããšæããŸãã
atanéæ¹é£ãã©ããã£ãŠäœã£ããããã¡ããšãŸãšããæ¹ãè¯ãæ°ããããã©ãã詳现ãèŠããŠãªããªãã
çµæ§çŽ°ããã¹ãããã«åããŠå°ããã€èšç®ããŠãã£ãããšã¯èŠããŠããã©ãã®ã¹ãããã®å解ã¯ããå¿åŽã®åœŒæ¹
ã®è¿äºããããŸããã
è€çŽ æ°z=(1+i)(2+i)âŠâŠ(n+i)ãçŽèæ°ãšãªãæ£ã®æŽæ°nããã¹ãŠæ±ããŠãã ããã
n=3 ã ãïŒ
äžè¬ã«arctan(1/n)ã«ã€ããŠã®æ§è³ªã調ã¹ãŠãããã次ã®ãããªé¢ä¿åŒãæç«ããŠããããšã«æ°ä»ããŸããã
arctan(1)-arctan(1/2)=arctan(1/3)
arctan(1)-arctan(1/3)=arctan(1/3)+arctan(1/7)
arctan(1)-arctan(1/4)=arctan(1/3)+arctan(1/7)+arctan(1/13)
arctan(1)-arctan(1/5)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)
arctan(1)-arctan(1/6)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)+arctan(1/31)
arctan(1)-arctan(1/7)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)+arctan(1/31)+arctan(1/43)
arctan(1)-arctan(1/8)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)+arctan(1/31)+arctan(1/43)+arctan(1/57)
arctan(1)-arctan(1/9)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)+arctan(1/31)+arctan(1/43)+arctan(1/57)+arctan(1/73)
arctan(1)-arctan(1/10)=arctan(1/3)+arctan(1/7)+arctan(1/13)+arctan(1/21)+arctan(1/31)+arctan(1/43)+arctan(1/57)+arctan(1/73)+arctan(1/91)
解çããããšãããããŸãã
n=3ã ããªæ°ãããŸããããããã蚌æãé£ããã§ããããã
Σarctan(1/(k^2+k+1)) =Σarctan(1/k)-arctan(1/(k+1)) =arctan(1/1)-arctan(1/(n+1)) ã§ããïŒ
ããšäžæ©ãŸã§è¿«ã£ãŠããæãã§ãããçŽæçã«ã¯èªæãªæåŸã®éšåãã©ã蚌æãããã®ãâŠâŠã
ãã®è€çŽ æ°ã¯ãå®éšèéšãšãæŽæ°ã§ãã
ãã£ãŠãæºèæ°ã§ãããªãã°ãã®è€çŽ æ°ã®çµ¶å¯Ÿå€ã¯èªç¶æ°ã§ãã
ãããã£ãŠã
â2 * â5 * â10 * âŠâŠ * â(n^2+1)
ãèªç¶æ°ã«ãªãããšãããªãã¡
2 * 5 * 10 * âŠâŠ * (n^2+1)
ãå¹³æ¹æ°ã«ãªãããšããå¿
èŠæ¡ä»¶ãšãªããŸãã
ãšããã§ãk^2+1 ãããçŽ æ° p ã®åæ°ã«ãªããããªèªç¶æ° k ã¯ã1âŠkâŠp-1 ã®ç¯å²ã«é«ã
2 ã€ãããªãã2 ã€ããå Žåã¯ãã®åã p ã«ãªããŸãã
ããªãã¡ãç©
2 * 5 * 10 * âŠâŠ * (n^2+1)
ã®äžã§ k^2+1 ãçŽ æ°ã§ããå Žåããããå¹³æ¹æ°ã«ãªãã«ã¯å°ãªããšã (k^2-k+1)^2+1 ãŸã§ç©ãç¶ããŠããå¿
èŠããããŸãã
ããŠãnâ§4 ã®è§£ããããã©ãããèããŸãã
4^2+1 = 17 ã¯çŽ æ°ã§ãã
ãã£ãŠãn ⧠17-4 = 13 ã§ããå¿
èŠããããŸãã
10^2+1 = 101 ã¯çŽ æ°ã§ãã
ãã£ãŠãn ⧠101-10 = 91 ã§ããå¿
èŠããããŸãã
90^2+1 = 8101 ã¯çŽ æ°ã§ãã
ãã£ãŠãn ⧠8101-90 = 8011 ã§ããå¿
èŠããããŸãã
ãããæéã®é£éã§æ¢ãŸãããšã nâ§4 ã§ãã解ãååšããå¿
èŠæ¡ä»¶ïŒååæ¡ä»¶ã§ã¯ãªãïŒã§ãã
ã€ãŸãã察å¶ãåãã°ããã®é£éãç¡éã«ç¶ãããšã瀺ãããã° nâ§4 ã«è§£ãååšããªã蚌æãšãªããŸãã
çŽæçã«ã¯èªæãªæããããŸããããã蚌æãããšãããããšãããŠã©ããããã®ãã
解çããããšãããããŸãã
ãªãã»ã©ïŒã€ãŸãn^2+1åçŽ æ°ãç¡éã«ååšããã°è¯ããšããããšã«ãªããŸãããããããããã¯ããã£ã³ãã¹ããŒäºæ³ãšããŠæªè§£æ±ºåé¡ã«ãªã£ãŠããããã§ãããããé£ããã
å°ãéããŸããã
n^2+1 åçŽ æ°ãç¡éã«ãã£ãŠãããã®é£éãç¡éã«ç¶ããšã¯éããŸããã
äŸãã°ïŒå®éã«ãããªããšã¯ãªããšæããŸããïŒã
ãã90^2+1 ã®æ¬¡ã«çŽ æ°ã«ãªãã®ã (10000ãè¶
ããæ°)^2+1 ã ã£ãå Žåãé£éãéåããŠãã9000ååŸã®ãšããã«è§£ãããå¯èœæ§ã¯æ®ããŸãã
ãŸããããã£ã³ãã¹ããŒã¯äžè¬çãªå€é
åŒã«ã€ããŠã®è©±ã§ããã
n^2+1ã«éã£ã話ã§ããã°ãã£ãšåçŽã«è§£æ±ºããå¯èœæ§ã¯ååã«ããã§ãããã
https://oeis.org/A101686
âãã¡ãã«ãããšããã®æ°åã§å¹³æ¹æ°ã¯1ãš100ã ããšèšŒæãããŠããããã§ãã
ãã£ãŠè§£ã¯n=3ã®ã¿ã§ããã
ããããšãããããŸãïŒãŸããã£ããèªãã§ã¿ãŸãïŒ
ããã§ã¯ããããåé¡ã§ãé¢çœããããããªãã§ãã
è€çŽ æ° z = (1^n + i)(2^n + i)(3^n + i)· · ·(k
^n + i) ãçŽèæ°ãšãªãæ£ã®æŽæ°ã®çµ (k, n) ãæ±ããŠãã ããã
(1^2+1)*(2^2+1)*(3^2+1)**(n^2+1)
ãå¹³æ¹æ°ãšãªãã®ã¯n=3ã®ã¿
ã«å¯Ÿã
(2^2-1)*(3^2-1)*(4^2-1)**(n^2-1)
ãå¹³æ¹æ°ãšãªãnã¯ïŒ
ãé¢çœãã£ãã§ãã
> (2^2-1)*(3^2-1)*(4^2-1)**(n^2-1) ãå¹³æ¹æ°ãšãªãnã¯ïŒ
n=((3+2â2)^(k+1)+(3-2â2)^(k+1)-2)/4ãïŒkã¯æ£æŽæ°ïŒ
ã§ããããã
äžè¬åŒã§äœãããã ïŒ
ãã¿ãªäžèŽããŠããŸãã
> è€çŽ æ° z = (1^n + i)(2^n + i)(3^n + i)· · ·(k
> ^n + i) ãçŽèæ°ãšãªãæ£ã®æŽæ°ã®çµ (k, n) ãæ±ããŠãã ããã
nâ§2 ã®å Žåã(2^n+i) 以éã®åè§ã®åèšã Ï/4 ã«å±ããŸããã
ãããã£ãŠç©ã®å®éšã¯åžžã«æ£ã§ãããçŽèæ°ã«ã¯ãªããŸããã
ãã£ãŠ n=1 ã®å Žåã®ã¿èããã°ãããå
ã®åé¡ã«åž°çããŸãã
2ç¹A(x1,y1),B(x2,y2)
ãéãçŽç·ã®æ¹çšåŒã
y-y1=(y2-y1)/(x2-x1)*(x-x1)
ã§äœ¿ãå
¬åŒããããããããè¡ååŒãå©çšããŠ
|x y 1|
|x1 y1 1|= 0
|x2 y2 1|
ãšãã圢åŒã«ããŠããã°
3ç¹A(x1,y1),B(x2,y2),C(x3,y3)ãéãåã®æ¹çšåŒã¯
|x^2 + y^2 x y 1|
|x1^2+y1^2 x1 y1 1|= 0
|x2^2+y2^2 x2 y2 1|
|x3^2+y3^2 x3 y3 1|
ãŸã空éã§ã
3ç¹A(x1,y1,z1),B(x2,y2,z2),C(x3,y3,z3)ãéãå¹³é¢ã®æ¹çšåŒã¯
|x y z 1|
|x1 y1 z1 1|= 0
|x2 y2 z2 1|
|x3 y3 z3 1|
åãã
4ç¹A(x1,y1,z1),B(x2,y2,z2),C(x3,y3,z3),D(x4,y4,z4)ãéãçé¢ã®æ¹çšåŒã¯
|x^2 + y^2 + z^2 x y z 1|
|x1^2+y1^2+z1^2 x1 y1 z1 1|
|x2^2+y2^2+z2^2 x2 y2 z2 1|= 0
|x3^2+y3^2+z3^2 x3 y3 z3 1|
|x4^2+y4^2+z4^2 x4 y4 z4 1|
ïŒå¿è«ååŸãæ£ã®å®æ°ã§ãšããããã«4ç¹ã¯éžã¶å¿
èŠã¯ãããŸãã
ãªã©ã§æ§æã§ããããã§ãã
ïŒå¹Ÿã€ãã§å®éšããã ãã§èšŒæããããã§ã¯ãããŸããã)
5ç¹A(x1,y1),B(x2,y2),C(x3,y3),D(x4,y4),E(x5,y5)ãéãåºçŸ©ã®äºæ¬¡æ²ç·(â»)ã®æ¹çšåŒã¯
|x^2 x*y y^2 x y 1|
|x1^2 x1*y1 y1^2 x1 y1 1|
|x2^2 x2*y2 y2^2 x2 y2 1| = 0
|x3^2 x3*y3 y3^2 x3 y3 1|
|x4^2 x4*y4 y4^2 x4 y4 1|
|x5^2 x5*y5 y5^2 x5 y5 1|
â»åºçŸ©ã®äºæ¬¡æ²ç·âŠãééåäºæ¬¡æ²ç·(æ¥åã»æŸç©ç·ã»åæ²ç·)ããã2çŽç·ããã1ç¹ããã1çŽç·ã
******
åå¿ç³»ããªããªã4ç¹A(x1,y1),B(x2,y2),C(x3,y3),D(x4,y4)ãéãåºçŸ©ã®çŽè§åæ²ç·(â»â»)ã®æ¹çšåŒã¯
|x^2 x*y y^2 x y 1|
|x1^2 x1*y1 y1^2 x1 y1 1|
|x2^2 x2*y2 y2^2 x2 y2 1| = 0
|x3^2 x3*y3 y3^2 x3 y3 1|
|x4^2 x4*y4 y4^2 x4 y4 1|
|1 0 1 0 0 0|
ãããã¯ãåŒå€åœ¢ããã°ã
|x^2-y^2 x*y x y 1|
|x1^2-y1^2 x1*y1 x1 y1 1|
|x2^2-y2^2 x2*y2 x2 y2 1| = 0
|x3^2-y3^2 x3*y3 x3 y3 1|
|x4^2-y4^2 x4*y4 x4 y4 1|
â»â»åºçŸ©ã®çŽè§åæ²ç·âŠãç矩ã®çŽè§åæ²ç·(挞è¿ç·ãçŽäº€ããåæ²ç·)ãããçŽäº€ãã2çŽç·ããã1çŽç·ã
ã¡ãªã¿ã«ã4ç¹A,B,C,Dãåå¿ç³»ããªãå ŽåãäžåŒã®å·ŠèŸºã¯(x,y)ã«äŸããæççã«0ã«ãªããŸãã
ãããæå³ããã®ã¯ãä»»æã®ç¹ã(4ç¹ãéã)åºçŸ©ã®çŽè§åæ²ç·äžã«ãããšããããšã§ãã
å®éã®ãšããã¯ãåå¿ç³»ããªã4ç¹ãéãåºçŸ©ã®çŽè§åæ²ç·ãç¡æ°ã«ååšãããã®4ç¹ãé€ãä»»æã®ç¹ã¯ãããã®ãã¡ã®1æ¬ã®äžã«ãããŸãã
******
GAIãããèŒããåã®æ¹çšåŒãã次ã®ããã«æžãã°äºæ¬¡æ²ç·ã«æ¡ä»¶ä»å ããããã®ãšããã®ããããããããªããŸãã
ãã ãã®åŒã¯è¡ååŒã®å±éãšåºæ¬å€åœ¢ã«ããç°¡åã«GAIããã®åŒã«ãªãã®ã§ãã¡ãªããã¯ããŸããããŸãããâŠâŠã
3ç¹A(x1,y1),B(x2,y2),C(x3,y3)ãéãåºçŸ©ã®å(â»â»â»)ã®æ¹çšåŒã¯
|x^2 x*y y^2 x y 1|
|x1^2 x1*y1 y1^2 x1 y1 1|
|x2^2 x2*y2 y2^2 x2 y2 1| = 0
|x3^2 x3*y3 y3^2 x3 y3 1|
|1 0 -1 0 0 0|
|0 1 0 0 0 0|
â»â»â»åºçŸ©ã®åâŠãç矩ã®åããã1çŽç·ã
å¥æ°ã¯ãå
šãŠãäžè¶³æ°ã®ããã§ãããéå°æ°ãããã®ã§ããããïŒ
å®å
šæ°ã¯ãèŠã€ãã£ãŠãªãããã§ããã
945
gp > sigma(945)-945
%485 = 975
ä»ã«ãå€ãã®éå°æ°ã¯å¥æ°ã®äžã«èŠã€ãããšæããŸãã
å§åçã«äžäœã®æ°ã¯5ã®ãã¿ãŒã³ãèµ·ãããããã®ã§ããïŒä»ã«ã
81081
153153
207207
189189
ãªã©ã®ããã§ããªããã®ãååšããŠããããã§ãã
GAIãããæ©éã®ãè¿äºããããšãããããŸãã
äºæ¡ã§ã¯ãèŠã€ãããŸããã§ããã
ç¡è¬ã«ããå¥æ°ã®å®å
šæ°ãããªãããšãèçæ³ã§ç€ºãããšããŠããŸããã
ïŒïŒ°ïŒŸÎ±âŠïŒ±ïŒŸÎ²ãå¥çŽ æ°ã®ç©ãšããŠã
ïŒïŒïŒâŠïŒïŒ°ïŒŸÎ±ïŒâŠïŒïŒïŒâŠïŒïŒ±ïŒŸÎ²ïŒïŒ2ããæãç«ã€ãšã
å³èŸºã¯ãçŽ å æ°ïŒããäžã€ã ããªã®ã§ã巊蟺ã®äžã€ããå¥æ°åãä»ã¯å¶æ°åã®çŽ å æ°ã«ãªãããšãããããŸããããå¥æ°åãšããŠã
αïŒïŒïœïŒïŒã®ãšãã¯ãïŒã®åæ°ã«ãªããççŸ
αïŒïŒïœïŒïŒã®ãšãã¯ãïŒïŒhïŒïŒãšãïŒhïŒïŒã«å ŽååããããŸãããããã以äžã¯é²ããŸããã§ããã
äžå以äžïŒïŒïŒïŒïŒïŒåïŒã®å¹³æ¹æ°ïŒéè€ãå¯ïŒã®åããŸãã¯ãå·®ã«ããã
ïŒïœïŒïŒïŒãŸã§ãè¡šãããšãã§ããŸããã
ïŒïŒïŒãã倧ããæ°ã«ã€ããŠã¯ãã©ãã§ããããïŒ
1,4,9,16,âŠã®é£æ¥é
ã®å·®ã3,5,7,âŠã§æ£ã®å¥æ°ã¯2å以äžã®å¹³æ¹æ°ã§è¡šããŸãã®ã§
ãããæ°ããå¥æ°ãªããã®ãŸãŸ2å以äžã§ãå¶æ°ãªãå¥æ°ã®å¹³æ¹æ°ã足ããåŒããããŠå¥æ°ã«ããããšã§
çµå±3å以äžã§è¡šããŸããã
ïŒè¿œèšïŒ
å¥æ°ã¯2å¹³æ¹æ°ã®å·®ã§è¡šããŸãããçµ±äžçã«3å¹³æ¹æ°ã®å æžã§è¡šãåŒãäœããŸããã
ä»»æã®æŽæ°nïŒè² ã®æ°ãå«ãïŒã«å¯ŸããŠ
(5[n/2]+17)^2-(10[n/2]+15-3n)^2-(4n+8-5[n/2])^2=n
ãæãç«ã¡ãŸãïŒ[ã]ã¯ã¬ãŠã¹èšå·ïŒã
ã¬ãŠã¹èšå·ã䜿ããã«
{(10n+63+5(-1)^n)/4}^2-{(4n+25+5(-1)^n)/2}^2-{(6n+37-5(-1)^n)/4}^2=n
ã®ããã«ãè¡šããŸãããå°ãé·ããªããŸãã
ïŒå{ã}å
ã¯æŽæ°ã«ãªããŸãïŒ
第2é
ã®ã«ãã³å
ã¯n=-5ã®ãšãã ã0ã第3é
ã®ã«ãã³å
ã¯n=-7ã®ãšãã ã0ã§ããã
6^2-5^2-4^2=-5, 1^2-2^2-2^2=-7ãæãç«ã€ããšããã
ãä»»æã®æŽæ°ã¯(èªç¶æ°)^2-(èªç¶æ°)^2-(èªç¶æ°)^2ã®åœ¢ã§è¡šããã
ããšãèšããŸãã
â»ç¬Šå·ãå転ããããšã§(èªç¶æ°)^2+(èªç¶æ°)^2-(èªç¶æ°)^2ã®åœ¢ã§ãè¡šããããšã«ãªããŸãã
æ¬æ¥ã®æ¥ä»(西æŠ2024幎3æ20æ¥) ã«ããããŠãå¹³æ¹æ°ãã€ãã¹å¹³æ¹æ°ãã€ãã¹å¹³æ¹æ°ã§è¡šèšã
50600817^2 -40480655^2 -30360488^2
= 20240320
ãªãã»ã©åŒ·ãâŠâŠâŠ
ããããããåã
ãªãã»ã©åãã
æçš¿åŸã«ããµãšæããŸãããã
50600817^2 -40480655^2 -30360488^2 = 20240320
ããã
5^2 -4^2 -3^2 = 0
ã®ãã¿ãŽã©ã¹ã®äžå¹³æ¹ã®å®çãæºããã5,4,3
ã®çµã¿ãã¡ãã£ãšããããŠããæããããŸããã
> 5,4,3ã®çµã¿ãã¡ãã£ãšããããŠããæããããŸããã
åŒã®äœãæ¹ããããŠçµæçã«ãããªããŸãã
çæ³ã¯
(am+b)^2-(cm+d)^2-(em+f)^2=2m or 2m+1
ãã
a^2-c^2-e^2=0
ab-cd-ef=1
b^2-d^2-f^2=0 or 1
ãšããæ¹çšåŒã解ãããšã§ãã
ããã°ã©ã ãäœã£ãŠæ¢çŽ¢ãããš
(5m+3)^2-(4m+2)^2-(3m+2)^2=2m+1
(5m+9)^2-(4m+8)^2-(3m+4)^2=2m+1
(5m+17)^2-(4m+12)^2-(3m+12)^2=2m+1
(5m+17)^2-(4m+15)^2-(3m+8)^2=2m
(5m+29)^2-(4m+21)^2-(3m+20)^2=2m
(5m+35)^2-(4m+30)^2-(3m+18)^2=2m+1
(13m+9)^2-(12m+8)^2-(5m+4)^2=2m+1
(13m+17)^2-(12m+15)^2-(5m+8)^2=2m
(13m+19)^2-(12m+18)^2-(5m+6)^2=2m+1
(13m+37)^2-(12m+35)^2-(5m+12)^2=2m
(17m+5)^2-(15m+4)^2-(8m+3)^2=2m
(17m+9)^2-(15m+8)^2-(8m+4)^2=2m+1
(17m+13)^2-(15m+12)^2-(8m+5)^2=2m
(17m+35)^2-(15m+30)^2-(8m+18)^2=2m+1
(25m+19)^2-(24m+18)^2-(7m+6)^2=2m+1
(25m+33)^2-(24m+32)^2-(7m+8)^2=2m+1
(25m+37)^2-(24m+35)^2-(7m+12)^2=2m
(29m+5)^2-(21m+4)^2-(20m+3)^2=2m
(29m+17)^2-(21m+12)^2-(20m+12)^2=2m+1
(37m+13)^2-(35m+12)^2-(12m+5)^2=2m
(37m+19)^2-(35m+18)^2-(12m+6)^2=2m+1
(37m+25)^2-(35m+24)^2-(12m+7)^2=2m
(41m+33)^2-(40m+32)^2-(9m+8)^2=2m+1
ã®ããã«ããããèŠã€ãããŸãããäžã«æžããåŒã¯æãç°¡åãª
(5m+17)^2-(4m+12)^2-(3m+12)^2=2m+1
(5m+17)^2-(4m+15)^2-(3m+8)^2=2m
ã®äºã€ãnã®å¶å¥ã©ã¡ãã§ãæãç«ã€ããã«
(5m+17)^2-(4m+27/2+(3/2)(-1)^n)-(3m+10-2(-1)^n)=n
ã®ããã«ãŸãšããmã[n/2]ã«ã(-1)^nã4[n/2]-2n+1ã«çœ®ãæããŠ
æŽçãããã®ãªã®ã§ãå€ã¯5ïŒ4ïŒ3ã«è¿ããªããŸãã
(13m+17)^2-(12m+15)^2-(5m+8)^2=2m
(13m+19)^2-(12m+18)^2-(5m+6)^2=2m+1
ã®äºã€ããŸãšããŠ
(13m+18-(-1)^n)^2-(12m+33/2-(3/2)(-1)^n)^2-(5m+7+(-1)^n)^2=n
ãšããŠæŽçããå Žåã¯
(9[n/2]+17+2n)^2-(6[n/2]+15+3n)^2-(9[n/2]+8-2n)^2=n
ãšããåŒã«ãªããããã«n=20240320ã代å
¥ãããš
131562097^2-121441935^2-50600808^2=20240320
ãšãªã£ãŠ13ïŒ12ïŒ5ã«è¿ããªããŸãã
ãããããããçŽ æŽãã解説ãæé£ãããããŸãã
3ä¹æ ¹[3]âã§è¡šããæ°å€ã«é¢ãã©ãããžã£ã³ã
(1) [3]â([3]â2 - 1) = [3]â(1/9) - [3]â(2/9) + [3]â(4/9)
(2) â([3]â5 - [3]â4) = ([3]â2 + [3]â20 - [3]â25)/3
ã®çåŒãèšããŠãããšã®èšäºã§èªã¿èšç®ãœããã§ç¢ºããããš
æ£ãããã¿ãªãšå³èŸº=巊蟺ãã®èšç®ãäžèŽããã§ã¯ãªããïŒ
gp > sqrtn(sqrtn(2,3)-1,3)
%233 = 0.63818582086064415301550365944406770127
gp > sqrtn(1/9,3)-sqrtn(2/9,3)+sqrtn(4/9,3)
%234 = 0.63818582086064415301550365944406770127
gp > sqrt(sqrtn(5,3)-sqrtn(4,3))
%235 = 0.35010697609230455692617090560659825895
gp > 1/3*(sqrtn(2,3)+sqrtn(20,3)-sqrtn(25,3))
%236 = 0.35010697609230455692617090560659825895
ãããæç«ããããšãè«ççã«ç€ºãã«ã¯ã©ãããããããã§ããããïŒ
èŠãéã3ä¹æ ¹ã§ã®çåŒã®å§¿ã¯æã£ãŠããªã圢ã§ç¹ãã£ãŠããŸããã§ããã
é¡ããçåŒãæãã€ããŸãããïŒ
(1)
1+t+t^2=(t^3-1)/(t-1)ã§t=[3]â2ãšãããš
1+[3]â2+[3]â4=1/([3]â2-1)
ãŸã
1-t+t^2=(t^3+1)/(t+1)ã§t=[3]â2ãšãããš
1-[3]â2+[3]â4=3/([3]â2+1)
ãã£ãŠ
{1-[3]â2+[3]â4}^3={3/([3]â2+1)}^3=27/(2+3[3]â4+3[3]â2+1)
=9/(1+[3]â2+[3]â4)=9([3]â2-1)
ãªã®ã§
[3]â(1/9)-[3]â(2/9)+[3]â(4/9)=[3]â([3]â2-1)
(2)
a=[3]â2, b=[3]â5ãšãããš
[3]â2+[3]â20-[3]â25=a+a^2b-b^2
(a+a^2b-b^2)^2
=a^2+a^4b^2+b^4+2a^3b-2ab^2-2a^2b^3
=a^2+2ab^2+5b+4b-2ab^2-10a^2
=9b-9a^2
=9([3]â5-[3]â4)
ãªã®ã§
([3]â2+[3]â20-[3]â25)/3=â([3]â5-[3]â4)
( A*x^2 + B*x*y + C*y^2 )^3 ã® A, B, C ãé©åœã«æ±ºãããã®ãçšæããŸãã
äŸãšããŠã( x^2 - x*y + y^2 )^3 ã§ãããŸãã
ãŸããå±éããŸãã
x^6 - 3*x^5*y + 6*x^4*y^2 - 7*x^3*y^3 + 6*x^2*y^4 - 3*x*y^5 + y^6
ææ°ã 3 ã§å²ã£ãããŸããçãããã®ããŸãšããŸãã
( x^6 - 7*x^3*y^3 + y^6 ) + x^2*y*( - 3*x^3 + 6*y^3 ) + x*y^2*( 6*x^3 - 3*y^3 )
ã©ããã® ( ) å
ã 0 ã«ãªãããã« x^3 ãš y^3 ã®å€ã決ããå
šãŠã® ( ) å
ã«ä»£å
¥ããŸãã
äŸãšããŠãx^2*y*( - 3*x^3 + 6*y^3 ) ã 0 ã«ãªãããã«ãx^3 = 2, y^3 = 1 ãšããŸãã
- 9 + 9*x*y^2
ããã§ã
( x^2 - x*y + y^2 )^3 = - 9 + 9*x*y^2
ãã§ããŸããã®ã§ã
x^2 - x*y + y^2 = ( - 9 + 9*x*y^2 )^(1/3)
ãåŸãããŸããã
æ®ã£ã x, y ã«ãäžä¹æ ¹ã®åœ¢ã§ä»£å
¥ãã䞡蟺 9^(1/3) ã§å²ãã°ã
[3]â([3]â2 - 1) = [3]â(1/9) - [3]â(2/9) + [3]â(4/9)
ãåŸãããŸãã
䌌ããããªæ¹æ³ã§åæ§ã®åŒããããã§ãäœããŸããã
DD++ããã®ã¢ããã€ã¹ã«ãã
(x^2-2*x*y+y^2)^3ã®å±éåŒãã
[3]â(25/9) - [3]â(80/9) + [3]â(4/9) = [3]â(7*[3]â(20) - 19)ãã®çåŒãçºç
gp > sqrtn(25/9,3)-sqrtn(80/9,3)+sqrtn(4/9,3)
%45 = 0.097375599902564072029769441954982002773
gp > sqrtn(7*sqrtn(20,3)-19,3)
%47 = 0.097375599902564072029769441954982004339
------------------------------------------------
(x^2-3*x*y+y^2)^3ã®å±éåŒãã
- [3]â(100) + [3]â(810) - [3]â(9) = [3]â(1241 - 273*[3]â(90)) ã®çåŒãçºç
gp > -sqrtn(100,3)+sqrtn(810,3)-sqrtn(9,3)
%52 = 2.6000248611968935936928541072898271257
gp > sqrtn(1241-273*sqrtn(90,3),3)
%53 = 2.6000248611968935936928541072898271256
-----------------------------------------------
(x^2-4*x*y+y^2)^3ã®å±éåŒãã
- [3]â(289/9) + 4*[3]â(68/9) - [3]â(16/9) = [3]â(631 - 91*[3]â(272)) ã®çåŒãçºç
gp > -sqrtn(289/9,3)+4*sqrtn(68/9,3)-sqrtn(16/9,3)
%56 = 3.4591342953019819946599609819643520211
gp > sqrtn(631-91*sqrtn(272,3),3)
%55 = 3.4591342953019819946599609819643520211
倩æã«ãªãããããªæèŠã«ãªããŸããã
調åæ°åãŒlognâγïŒãªã€ã©ãŒïŒ
äžå®åœ¢âïŒâã®åœ¢ãããŠããŸãã
ä»ã«ãïœæ¬¡å€é
åŒãŒïœæ¬¡å€é
åŒ
lim(ïœââ)ïœlog(ax^n+ ⊠)-log(bx^n+âŠãã)ïœ=loga/b
ãã®ä»ããããŸãããããææãã ããã
â(n+3ân) - â(n-ân)â2
(n!)^(1/n) - (n-1)!^(1/(n-1))â1/e
ãªã©ãèµ·ããããã§ãã
ã¡ãã£ãšè©±é¡ããããŸãã
â - â
ã®ããŒããšããŠãææ°žæ¯äžéå
çã»ããããŒãã«è³ãããã£ãããããã¿çè«ããšããã®ãæãåºããŸããã
埡åè: http://catbirdtt.web.fc2.com/kurikomirironntohananika.html
Dengan kesaktian Indukmuããã玹ä»ããããªã³ã¯ãèªãã§ã¿ãŠãããã«åºãŠãã137ãšããçŽ æ°ã«
é
ãããã人ç©ã«ãã¡ã€ã³ãã³ãšããŠãªãæãåºããŸãã
æã«ããŠãªã¯è¥ãããŠ(58æ³)èµèçã§äº¡ããªã£ããšãããããã®æã«å
¥é¢ããŠããéšå±ã®çªå·ãæ£ã«
èå³ãæ±ãç¶ããŠããæ°å€ã«ãã¿ãªäžèŽãã137å·å®€ã§ãã£ãããšã«ãèªåã®éåœãå¯ç¥ãããšãã
éžè©±ãäŒããããŠãããšããã
æ°åŠè
ãããã§ãããç©çåŠè
ãã»ããšã«äºçŽ°ãªããšã«çŽ°å¿ã®æ³šæãæãèåŸã«æœãé¢ä¿æ§ãæ³åã
ãã®ã®èŠäºã«æŽãåãå端ãªãã§ããã
ããŒãšçããŠãããããŒããšãã³ã¡ããã«å±ããããã§ãã
(ΣïŒ/k)^2ïŒÎ£ïŒ/ïœïœïŒïœãšïœã¯ç°ãªãïŒïŒÎ£ïŒ/n^2
ããããç¡éã«èšç®ãããšãã©ãã§ããããïŒ
âïŒn^2+2nïŒãŒãn ãâãïŒ