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评论
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猜想:(i)n(n)>0,n=0,a(n)=1,n=7, 39, 47,95, 191, 239,327, 439, 871,1167, 1199, 1367,1487, 1727, 1751,2063, 2351, 2471,4647, 4,k*m(k=0,1,2,…)m=1, 3)。
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (a*x+b*y+c*z)*x*y*z is a square, whenever (a,b,c) is among the triples (1,3,7), (1,5,7), (1,5,11), (1,13,23), (2,4,6), (2,4,8), (2,6,8), (2,8,26), (3,5,21), (3,7,15), (3,9,43), (3,9,69), (3,9,141), (3,21,27), (3,27,39), (3,33,45), (3,39,123), (6,8,12), (6,8,18), (6,8,22), (6,8,28), (6,12,48), (6,18,132), (6,24,34), (6,24,36), (6,42,72), (7,13,29), (7,19,23), (12,18,24), (12,18,30), (12,26,48), (13,15,21), (13,17,19), (13,33,39), (14,28,58), (15,45,51), (16,22,62), (18,22,24), (21,27,33), (21,27,57), (23,37,61), (24,54,66), (33,57,79), (38,48,66), (42,58,84), (46,92,118).
对于拉格朗日四方定理的进一步改进,参见ARXIV:1604.06723。
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