The recently developed proof of Fermat's Last Theorem is very lengthy and difficult, so much so as to be beyond all but a small body of specialists. While certainly of value in the developments that resulted, that proof could not be, nor was offered as being, possibly the proof Fermat had in mind. The present proof being brief, direct and concise is a candidate for being what Fermat had in mind. It is also completely accessible to any one trained in common algebra. That critical suggestions offered by significant mathematics authorities have been unable to invalidate this concise and direct proof would tend to be major confirmation that: The proof stands, valid and not validly challenged.
费马大定理（Fermat's Last Theorem，FLT）指出，方程$x^n+y^n=z^n$对于任何整数指数$n>2$都没有正整数$x, y, z$的解。但是这个定理有一个定量的版本吗？在最初的调查中，我们发现了一个无穷序列的整数$(x_n, y_n, z_n)$与$x_n^4+y_n^4-8=z_n^2$。
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。
