84 Proving Fermat’s Last Theorem

It was the most tantalizing statement in higher mathematics, and it had been hastily scribbled into the margin of a book. Elementary school students learn the Pythagorean theorem, which is related to it, that a² + b² = c². The statement in the margin, however, was a negative, that a³ + b³ does not equal c³, and that, furthermore, the equation is just as invalid in any situation where the exponent is above 2. So a⁴ + b⁴ never equals c⁴, a⁵ + b⁵ never equals c⁵, and so on.

The marginal statement had been left by eminent 17th-century mathematician Pierre de Fermat, and it contended that he had discovered a proof for his conjecture—but that there was not enough room to write it in the margin. Fermat’s statement intrigued and challenged mathematicians for the next 300 years, for no one could prove it. In higher mathematics, it is not sufficient merely to state that a theorem is true even if it holds true for every number that you try it with; you must devise a proof that can be demonstrated to work with any possible number inserted into the equation. A statement has little or no value until such a proof is substantiated.

Fermat’s Last Theorem, as it was called, was the most significant unproved theorem in higher mathematics, and it was not conclusively demonstrated to be true until 1994. At that time, mathematician Andrew Wiles, who had spent years struggling with the problem, corrected his earlier 1993 proof, and his astonished colleagues certified his work as legitimate.

At 150 pages, however, Wiles’s proof is certainly not the same one Fermat envisioned centuries ago. In that respect, Fermat’s marginal note will remain an enigma forever.

84 证明费马大定理

这是高等数学中最诱人的陈述，它被匆忙地写在一本书的页边空白处。小学生会学习与之相关的毕达哥拉斯定理：a² + b² = c²。但页边的陈述是否定的——a³ + b³不等于c³，而且，当指数大于2时，这个等式在任何情况下都是不成立的。所以a⁴ + b⁴永远不等于c⁴，a⁵ + b⁵永远不等于c⁵，依此类推。

这条页边陈述是17世纪著名数学家皮埃尔·德·费马留下的，他声称自己已经发现了这个猜想的证明——但页边空白处没有足够的空间写下证明过程。费马的陈述在接下来的300年里吸引并挑战着数学家们，因为没人能证明它。在高等数学中，仅仅声明一个定理是正确的是不够的，哪怕它对所有你尝试过的数字都成立；你必须设计出一个证明，能证明这个等式对任何可能代入的数字都成立。在这样的证明得到证实之前，这个陈述几乎没有价值。

这个被称为“费马大定理”的命题，是高等数学中最重要的未被证明的定理，直到1994年才被最终证明为真。当时，数学家安德鲁·怀尔斯（他已花费数年时间研究这个问题）修正了他1993年的早期证明，他震惊的同事们证实了他的工作是合法有效的。

然而，怀尔斯的证明长达150页，显然不是费马数百年前设想的那个证明。从这个角度来说，费马的页边注释将永远是一个谜。