丢番图方程x^3/3+y^3+z^3-2xyz=0
The diophantine equation x^3/3+y^3+z^3-2xyz=0
摘要：
在本文中我们将提出两个定理。第一个定理是关于三次丢番图方程不存在整数解的一个新结果。在这个定理的证明中，我们使用了二元三次型理论和无穷下降法中的一些已知结果，这些结果在初等数论（ENT）的范围内是很好理解的。在第二个定理中，我们证明了著名的费马最后定理（FLT）的指数为3和第一个定理是等价的。因此定理1和定理2构成了这个著名的三次费马方程不存在整数解的另一个证明。众所周知，从L.Euler（1770）到F. J. Duarte（1944），许多人已经给出了指数为3的FLT的证明。但是所有的证明都使用了概念，这超出了ENT的范围。因此，与其他证明不同，这里给出的证明是FLT的ENT证明，指数为3。
英文摘要：
We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from theory of binary cubic forms and the method of infinite descent, which are well understood in the purview of Elementary Number Theory(ENT). In the second theorem, we show, that the famous Fermat's Last Theorem(FLT) for exponent 3 and the first theorem are equivalent. So Theorem1 and 2 constitute an alternate proof for the non-existence of integer solutions of this famous cubic Fermat's equation. It is well known, that from L.Euler(1770) to F.J.Duarte(1944) many had given proof of FLT for exponent 3. But all proofs uses concepts, which are beyond the scope of ENT. Hence unlike other proofs the proof given here is as an ENT proof of FLT for exponent 3.
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