## 2012年9月19日 星期三

### 公元4世紀的莎草紙.......，望月新一可望證明abc猜想 Diophantine equations

〔編 譯俞智敏／綜合報導〕專研基督教歷史的哈佛大學神學院教授金恩（Karen King），18日公布一份公元4世紀的莎草紙，上面用古埃及科普特文字引述耶穌明確提到他的「妻子」。這項發現恐在基督教世界再度引發耶穌是否結婚的激 烈辯論，甚至可能推翻基督教崇尚禁欲的理想。

〔編譯林翠儀／綜合報導〕日本天才數學家、京都大學數學教授望月新一，發 表一篇長達500頁的論文，可望證明被提出將近30年的「abc猜想」（abc conjecture），如果證明正確，將可能在數論領域掀起一場革命，也將成為本世紀最偉大的數學成就。望月的證明目前仍在進行檢證中。
「abc 猜想」是由歐洲數學家馬瑟（David Masser ）與奧斯達利（Joseph Oesterle）在1985年分別提出，以a和b兩個整數及其相加後新的整數c，abc三個數組與各個質數之間的關係，可說是一個針對質數之間深層聯繫 的猜想。近30年來，許多數學家都想證明這個猜想，荷蘭萊頓大學數學研究所甚至成立一個網站計算平台「ABC@Home」，協助數學家破解這個猜想。
500頁論文 花10年才完成

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In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it.
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems Diophantus initiated is now called "Diophantine analysis". A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.
While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth century.

## Examples of Diophantine equations

 In the following Diophantine equations, x, y, and z are the unknowns, the other letters being given are constants. $ax+by=1\,$ This is a linear Diophantine equation (see the section "Linear Diophantine equations" below). $x^n+y^n=z^n \,$ For n = 2 there are infinitely many solutions (x,y,z): the Pythagorean triples. For larger values of n, Fermat's Last Theorem states there are no positive integer solutions (x, y, z). $x^2-ny^2=\pm 1\,$ (Pell's equation) which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century. $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution in x, y, and z, all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4xyz = yzn + xzn + xyn = n(yz + xz + xy).