摘要
本文证明了,若a、b、c和d为全实数域k中的元素且其中至少有一个是全正数,则k中每个非负元r均可表为x2+y2+z2+w2使得x、y、z和w为k中非负元并且ax+by+cz+dw为k中的平方元.若将ax+by+cz+dw为k中的平方元换为ax+by+cz+dw为k中的立方元,结论也成立.这是孙智伟的一个猜想.
Abstract
We prove one of Sun Zhi-Wei's conjectures.Let k be a totally real field and set k≥0={t ∈ k:t≥0}.Let a,b,c,d ∈ k,where one of all is totally positive and n ∈ {2,3}.Then each r ∈ k≥0 can be written as x2+y2+z2+w2 with x,y,z,w ∈ k≥0 such that ax+by+cz+dw ∈ {tn:t ∈ k}.
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