ïŒïŒïŒïŒa^2+b^2+c^2+d^2
äœããabcd=ïŒã§ãªããè§£ãæ±ãã
ïŒïŒïœâŠïœâŠïœâŠïœããšããŠã
ïŒïœïŒïœïŒïœïŒïœïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒãïŒïŒïŒïŒïŒïŒïŒïŒïŒãïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒãïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
0ïŒaâŠbâŠcâŠd ãšããŠ
(a,b,c,d)=(1,1,2,12),(1,2,8,9),(1,6,7,8),(2,3,4,11),(2,4,7,9),(3,4,5,10),(4,6,7,7),(5,5,6,8)
æŽæ°ã®å¶éãä»ããŠãªãã®ã§ä»ã«ã
(14/5,33/5,47/5,16/5)
(33/13,79/13,112/13,74/13)
(56/25,137/25,193/25,186/25)
(41/21,103/21,48/7,26/3)
(48/23,119/23,238/23,79/23)
(109/57,275/57,550/57,104/19)
(12/7,31/7,62/7,7)
(52/33,137/33,359/33,38/11)
(115/79,309/79,812/79,410/79)
(6/5,17/5,56/5,17/5)
(31/17,127/17,158/17,36/17)
(53/31,221/31,274/31,132/31)
(39/25,167/25,206/25,148/25)
(101/71,445/71,718/71,180/71)
(4/3,6,29/3,13/3)
(21/19,101/19,202/19,52/19)
(16/13,103/13,119/13,18/13)
(107/91,701/91,808/91,42/13)
(50/51,353/51,168/17,95/51)
(31/37,303/37,334/37,32/37)
(119/27,64/9,238/27,41/27)
(55/13,89/13,110/13,48/13)
(155/39,84/13,310/39,211/39)
(137/37,224/37,361/37,78/37)
(103/29,169/29,272/29,114/29)
(153/49,254/49,508/49,17/7)
(271/81,623/81,718/81,28/27)
(153/47,353/47,406/47,136/47)
(101/35,237/35,338/35,8/5)
(149/57,458/57,168/19,37/57)
(224/43,313/43,359/43,18/43)
(169/33,709/99,812/99,226/99)
(254/55,357/55,508/55,61/55)
(505/119,921/119,1010/119,4/17)
(17/3,22/3,8,1/3)

ä»ã«
(3,5,11,131)
(3,5,29,113)
(3,5,41,101)
(3,5,53,89)
(3,5,59,83)

(11,19,47,73)
(11,19,53,67)
(11,19,59,61)
(11,23,37,79)
(11,23,43,73)

(19,29,31,71)
(19,29,41,61)
(19,29,43,59)
(19,31,41,59)
(19,31,47,53)
(19,37,41,53)
(19,41,43,47)
(23,29,31,67)
(23,29,37,61)
(23,31,37,59)
(23,31,43,53)
(23,37,43,47)
(29,31,37,53)
(29,31,43,47)
(29,37,41,43)

ãã ãæ°å€ã«âèšå·ãä»ããŠäžããã
æŽã«iãèæ°åäœãšã
(-10+i,-9+5*i,-5+7*i,10+9*i)
(-10+3*i,-9+4*i,2+5*i,8+7*i)
(-10+6*i,-8+7*i,-6+10*i,16+11*i)
(-9+2*i,-8+4*i,-5+6*i,10+8*i)
(-8+i,-6+2*i,-4+3*i,8+4*i)
(-8+3*i,-6+5*i,-5+6*i,12+7*i)

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é¢çœããªãåçã§ããã
宿°ç¯å²ã®äžè¬è§£ã¯p,q,rãä»»æã®æ£ã®å®æ°ãšããŠ
(a,b,c,d)=(屉{150/(1+p+q+r)},屉{150p/(1+p+q+r)},屉{150q/(1+p+q+r)},屉{150r/(1+p+q+r)})
è€çŽ æ°ç¯å²ã®äžè¬è§£ã¯p,q,rãä»»æã®è€çŽ æ°ïŒãã ãpqrâ 0ãã€p^2+q^2+r^2â 150ïŒãšããŠ
(a,b,c,d)=(p,q,r,屉(150-p^2-q^2-r^2))
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ïŒïŒïŒŸïŒïŒïŒïŒŸïŒïŒïŒïŒŸïŒïŒïŒïŒŸïŒ
ïŒïŒïŒŸïŒïŒïŒïŒiïŒ^2+ïŒïŒiïŒ+i^2
14^2+(6i)^2+(3i)^2+i^2
è€çŽ æ°ã«ã¯å€§å°é¢ä¿ã¯ãããŸãããã
ãããããïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒãŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒãšïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
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ïŒa,b,c,d)ã®ïŒïŒïŒïŒïŒä¹åãŸã§ãçããã®ãèŠã€ãããš
ïŒ-a,-b,-c,-d,a,b,c,d)ããïŒåã®çޝä¹åãèŠã€ããäœæŠã§ãã
2(1,2,3,4)+(a,b,c,d)
äœããa,b,c,d ã¯ãé åºé¢ä¿ãªã1,2,3,4笊å·ãã€ãããã®ã§ãã
2(1,2,3,4)+(-4,-1,2,-3)=(-2,3,8,5)
2(1,2,3,4)+(-2,3,-4,-1)=(0,7,2,7)
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