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(1)
a^2+b^2=c^2+d^2=m1, aïŒcïŒdïŒbãšããŠ
èªç¶æ°ã®å¹³æ¹æ°ã®éå·®ã¯3,5,7,9,11,13,15,âŠã§ãã3+5+7=15ãªã®ã§
c^2-a^2=3+5+7ããa^2=1,c^2=16ãb^2-d^2=15ããb^2=64,d^2=49ã€ãŸã
1^2+8^2=4^2+7^2=65ãããã«èŠã€ãããŸãã
èãèœãšãããªããã°ãããæå°ã ãšæããŸãã
(2)
(1)ãšåæ§ã«
a^2+b^2=c^2+d^2=e^2+f^2=m2, aïŒcïŒeïŒfïŒdïŒbãšèããŸãã
mod6ããçŽ æ°ã¯5以äžã®å¥æ°ã§ãããã
5以äžã®å¥æ°ã®å¹³æ¹æ°ã®éå·®24,32,40,48,56,âŠã®1/8ããªãã¡
3,4,5,6,7,âŠããc^2-a^2=b^2-d^2,e^2-c^2=d^2-f^2ãšãªããã®ãæ¢ããŸãã
ãã ãçŽ æ°ã§ãªãå¥æ°ãå«ãŸããŠããããšã«ãçæããŸãã
3+4=7âc=9ã§åææ°ãªã®ã§äžé©
3+4+5=12âa=5,c=11,b=25,d=23ãšãªãbãåææ°ãªã®ã§äžé©
â»ã¡ãªã¿ã«äž¡èŸºã®ç«¯ã®æ°ã«ãã®é£ã®æ°ãå ç®ããã°a,b,c,dã¯ããã«åºãŸãã
â»3+4+5=12ã§ã¯a=2+3=5,c=5+6=11,b=12+13=25,d=11+12=23ãšãªããŸãã
4+5=9âa=7,c=11,b=19,d=17ãšãªãããéã6,7,8ãããªãã®ã§äžé©
ïŒå€åçŽ æ°ã§2éããªã7^2+19^2=11^2+17^2=410ãæå°ã ãšæããŸãïŒ
3+4+5+6=18âa=5,c=13,b=37,d=35ãšãªãdãåææ°ãªã®ã§äžé©
4+5+6=15âa=7,c=13,b=31,d=29
ãã®ãšãéã7,8,9,10,11,12,13,14ãªã®ã§
14ïŒ7+8ïŒ7+8+9ïŒ14+13ïŒ7+8+9+10ïŒ14+13+12ïŒ7+8+9+10+11
ããäžé©
5+6=11âa=9ã§åææ°ãªã®ã§äžé©
3+4+5+6+7=25âc=15ã§äžé©(åæ§ã«+7ã§çµãããã®ã¯ãã¹ãŠäžé©)
3+4+5+6+7+8=33âa=5,c=17,b=67,d=65ãšãªãdãåææ°ãªã®ã§äžé©
3+4+5+6+7+8=11+12âb=25ã§äžé©(éã9,10ãããªãã®ã§ããããç¡ç)
4+5+6+7+8=30âa=7,c=17,b=61,d=59
ãã®ãšãéã9,10,11,âŠ,29ãªã®ã§
9+10ïŒ29ïŒ9+10+11ïŒ9+10+11+12+13ïŒ28+29ïŒ9+10+11+12+13+14
ïŒ9+10+11+12+13+14+15=27+28+29ãã
e=31,f=53ã§æãç«ã€
ã€ãŸã7^2+61^2=17^2+59^2=31^2+53^2=3770
ãããæå°ãã©ããããããªãã®ã§5+6+7+8,6+7+8,7+8ãèŠæ€èš
5+6+7+8=26âa=9ã§äžé©
6+7+8=21âa=11,c=17,b=43,d=41
ãã®ãšãéã9,10,11,âŠ,20ã ã
9+10ïŒ20ïŒ9+10+11ïŒ19+20ïŒ9+10+11+12ïŒ9+10+11+12+13ïŒ18+19+20
ïŒ9+10+11+12+13+14ïŒ17+18+19+20ïŒ9+10+11+12+13+14+15ïŒ16+17+18+19+20
ãªã®ã§äžé©
7+8=15âa=13,c=17,b=31,d=29
ãã®ãšãéã9,10,11,12,13,14ã ã
14ïŒ9+10ïŒ13+14ïŒ9+10+11ïŒ12+13+14ãªã®ã§äžé©
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èãèœãšãããªããã°7^2+61^2=17^2+59^2=31^2+53^2=3770
(3),(4)ã¯å€§å€ãããªã®ã§ãšãããããã¹ã
No.532ãããã2023幎2æ21æ¥ 03:19
ç§ã«ã¯åœç¶ã©ããèŠèœãšããªã®ãå€æã§ããŸãããã
29^2+37^2=23^2+41^2=19^2+43^2(=2210)
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(1)ãªã©ã®è§£æ³ã¯æã£ãŠãããŸããã§ããã
No.533GAI2023幎2æ21æ¥ 07:32
ãªãã»ã©ãäžèšã§4+5+6+7+8=30ãå
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å³èŸºã®æ倧å€ã30æªæºã®ãã®ã¯ããã«æ€èšããå¿
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çããããã£ãŠããŸãã®ã§ç¶ããæ€èšããã®ã¯ãããŸããã
19^2+43^2=23^2+41^2=29^2+37^2ãšããããšã¯
10+11=21
12+13+14=19+20
ãã
a=9+10=19, c=11+12=23, b=21+22=43, d=20+21=41,
e=14+15=29, f=18+19=37
ãšãªãããã§ããã
ã¡ãªã¿ã«ãç§ãæžãã3770ã¯2çªç®ã«å°ãã解ã§ããã
ãããããããã°ã©ã ãäœã£ãŠã¿ãŸããã
(1)ã®æå°è§£ã¯1^2+7^2=5^2+5^2=50ã§ããã
(3)ã¯âããã«ããããã«635318657ã§ããã
https://oeis.org/A046881
No.534ãããã2023幎2æ21æ¥ 09:39
(1) ã® m1 㯠5 ã®åæ°ã§ããïŒ
ãšããã¯ã€ãºãâŠâŠ
50 = 1^2 + 7^2 = 5^2 + 5^2
65 = 1^2 + 8^2 = 4^2 + 7^2
85 = 2^2 + 9^2 = 6^2 + 7^2
125 = 2^2 + 11^2 = 5^2 + 10^2
130 = 3^2 + 11^2 = 7^2 + 9^2
145 = 1^2 + 12^2 = 8^2 + 9^2
170 = 1^2 + 13^2 = 7^2 + 11^2
185 = 4^2 + 13^2 = 8^2 + 11^2
205 = 3^2 + 14^2 = 6^2 + 13^2
No.535Dengan kesaktian Indukmu2023幎2æ21æ¥ 17:12
11^2+10^2=14^2+5^2=221
29^2+11^2=31^2+1^2=962
18^2+13^2=22^2+3^2=493
23^2+15^2=27^2+5^2=754
25^2+19^2=31^2+5^2=986
16^2+11^2=19^2+4^2=377
27^2+23^2=33^2+13^2=1258
23^2+10^2=25^2+2^2=629
äœã§ãããã§ããã
No.536GAI2023幎2æ21æ¥ 18:12
221,962,493,754,986,377,1258,629ã¯ãã¹ãŠ
4n+1åã®çžç°ãªãçŽ å æ°ãäºã€æã€æ°ã§ããã
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p=a^2+b^2, q=c^2+d^2 ã®ãšã
pq=(ad+bc)^2+(ac-bd)^2=(ad-bc)^2+(ac+bd)^2
ãšãªãããã§ãã
ãããç¹°ãè¿ãã°ãçžç°ãªãmåã®4n+1åã®çŽ æ°ã®ç©ã2å¹³æ¹åã§è¡šãæ¹æ³ã¯
2^(m-1)éããšããããŸããã
äŸ
5Ã13Ã17=1105=4^2+33^2=9^2+32^2=12^2+31^2=23^2+24^2
ã¡ãªã¿ã«äžèšã®æ°ã®ãã¡å¶æ°ã®ãã®ã¯çžç°ãªãäºã€ã®4n+1åã®çŽ æ°ã®ç©ã®2åã§ããã
p=a^2+b^2 â 2p=(a+b)^2+(a-b)^2 ããã2åããŠãè¡šãæ¹æ³ã¯å¢ãããæžã£ããããŸããã
äŸãã°962=2Ã13Ã37ã¯
13=2^2+3^2, 37=1^2+6^2ãªã®ã§
13Ã37=(2Ã6+3Ã1)^2+(2Ã1-3Ã6)^2=15^2+16^2ãã
2Ã13Ã37=(15-16)^2+(15+16)^2=1^2+31^2
13Ã37=(2Ã6-3Ã1)^2+(2Ã1+3Ã6)^2=9^2+20^2ãã
2Ã13Ã37=(9-20)^2+(9+20)^2=11^2+29^2
ã®ããã«æ©æ¢°çã«ç®åºã§ããŸãã
No.537ãããã2023幎2æ22æ¥ 16:06
> çžç°ãªãmåã®4n+1åã®çŽ æ°ã®ç©ã2å¹³æ¹åã§è¡šãæ¹æ³ã¯2^(m-1)éããšããããŸããã
ã»ãã®ãããªæ¹æ³ã§äœã£ã 2^(m-1) éãã®ãªãã«éè€ããªãããš
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No.538DD++2023幎2æ23æ¥ 08:53