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评论
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猜想:(i)a(n)>0,n=0,a(n)=1,仅n=4 ^ k*m(k=0,1,2,…)m=1, 7, 23、31, 39, 47、55, 71, 79、119, 151, 191、311, 671)。
(ii)任何自然数都可以写成x^ 2+y^ 2+z ^ 2+w ^ 2(x+y+z)^ 2+(4×(x+yz))2平方,其中x,y,z,w是x+y>=z的非负整数。
(iii) For each tuple (a,b,c,d,e,f) = (1,1,1,3,6,-3), (1,1,1,4,12,-12), (1,1,2,1,1,-5), (1,1,2,1,8,-5), (1,1,2,3,3,-3), (1,1,2,4,4,-8), (1,3,11,12,4,4), (1,3,14,16,4,4), (1,3,14,18,4,2), (1,3,20,16,4,12), (1,4,11,6,3,3), (1,5,13,12,12,12), (1,5,14,15,12,21), (1,6,6,16,8,8), (1,6,14,12,8,8), (1,6,14,16,8,4), (1,6,17,20,8,4), (1,6,20,20,8,8), (1,7,8,4,2,6), (1,7,8,10,5,15), (1,7,9,10,5,12), (1,7,15,4,2,8), (1,7,15,10,5,20), 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)^2 + (d*x+e*y+f*z)^2 is a square.
在ARXIV中证明:1604.06723,任何正整数都可以被写为x ^ 2 +y^ 2 +z ^ 2 +w ^ 2,其中x,y,z,w非负整数和y> 0,使得x+4*y+4*z和9×x+3*y+3*z是正整数边的右三角形的两条腿。
也见A171714,A73107,A73108和A73134对于与毕达哥拉斯三元组有关的猜想。对于拉格朗日四方定理的更多猜想,可以参考ARXIV:1604.06723。
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