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楼主
发表于 2012-9-19 16:51
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[转帖] 声称两个质数之间的深度联系的证明
http://www.ecocn.org/thread-82852-1-1.html
[导度] 看看没有人选, 就捷足先登,试试. 没想到文章有那么多的谬误. 明显大都是排版问题, 但也不乏作者罗嗦之嫌. 比如 sqp(abc)^r/c,其中r= 2，Sqp(abc)^r/c是几乎总是大于1，和总是大于零。
该说的没有谈及. 怀疑自然杂志的质量在下降. 也不乏我们的水平不够. 认识有误. 请数学家, 爱好者进一步指正.
Proof Claimed for Deep Connection between Prime Numbers
声称两个质数之间的深度联系的证明
If true, a solution to the "abc" conjecture about whole numbers would be "one of the most astounding achievements of mathematics of the 21st century
如果为真，解决“ABC”猜想将是“21世纪最惊人的数学成就之一
时间:2012-09-1717时47分来源：环球科学（huanqiukexue.com）
From Nature magazine 
The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.
Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof of the abc conjecture, which proposes a relationship between whole numbers — a 'Diophantine' problem.
“已经解决了数论中的最重要的问题之一”，这一则声明使得往日里平静的数学界热闹了。
日本京都大学的数学家真一望月发布了一份500页的ABC猜想的证明，提出了两个整数之间的关系 ---“丢番图”的问题的解释。
The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem,” says Dorian Goldfeld, a mathematician at Columbia University in New York. “If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.”
ABC猜想是大卫.马瑟尔和约瑟夫.斯蒂芬欧斯特在1985年分别独立提出的，对这个世界来说，它可能不如费马大定理那么令人知晓，但在某些方面，它更为重要。“如果证明ABC猜想是正确的，就一举解决了许多著名的丢番图的问题，其中包括费马大定理”，纽约哥伦比亚大学的数学家多利安.戈德菲尔德说，“如果望月的证明是正确的，它将是二十一世纪数学的最令人瞩目的成就之一。”
Like Fermat’s theorem, the abc conjecture refers to equations of the form a+b=c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers [**应为15 and 17 are square-free numbers，估计排版错误，译者注], but 16 and 18 — being divisible by 42 and 32, respectively — are not.
The 'square-free' part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)=2×3=6.
如费马大定理一样，ABC猜想是指形式为A + B= C的方程。它涉及一个无平方数的概念：一个不能被划分成任何数的平方的数字。15和17是无平方数，但16和18不是，它们可以分别被42和32除。
///从后面的译者注释看，这里应当是4^2; 3^2才对。译者注///
一个数字n的“无平方”部分，sqp(n), 是最大的无平方数，它可以用n的诸因数相乘构成，这些因数是n的质数（素数）。比如，sqp（18）= 2×3= 6。
If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers axbxc, or abc — or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, the ratio of sqp(abc)r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)2/c = 900/128. In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero.
如果你明白了这一点，那么你懂得abc猜想了。它涉及三个整数 axbxc 或 abc 的积的性质 - 更具体些，涉及到这个积的无平方部分，包括其不同的质因数。它指出，对于整数a+b=c, 对于大于1的任意r值，sqp(abc)r/c总是有大于零的某个最小值。例如，如果a= 3 和 b= 125，则c= 128，那么sqp（abc）=30和sqp（abc）2/c= 900/128。在这种情况下，其中r= 2，Sqp(abc)r/c是几乎总是大于1，和总是大于零。
/// sqp（abc）2/c= 900/128，从数字看，应当是[sqp（abc）^2]/c= 900/128. 同样sqp(abc)r/c为sqp(abc)^r/c。译者注。///
Deep connection
It turns out that this conjecture encapsulates many other Diophantine problems, including Fermat’s Last Theorem (which states that an+bn=cn has no integer solutions if n>2). Like many Diophantine problems, it is all about the relationships between prime numbers. According to Brian Conrad of Stanford University in California, “it encodes a deep connection between the prime factors of a, b and a+b”.
深度连接
事实证明，这个猜想囊括了许多其他的丢番图问题，包括费马大定理（即a^n+b^n=c^n当n> 2时，没有整数解）
/// an+bn=cn显然排错版了，译者注///。
像许多丢番图问题一样，它全是关于两个质数之间的关系。按加利福尼亚州斯坦福大学的布赖恩•康拉德的话，“它勾勒了a, b所有的质因数和a+b之间的深刻连接”。
Many mathematicians have expended a great deal of effort trying to prove the conjecture. In 2007, French mathematician Lucien Szpiro, whose work in 1978 led to the abc conjecture in the first place claimed to have a proof of it, but it was soon found to be flawed.
许多数学家们费了大量的努力试图证明这个猜想。法国数学家Lucien Szpiro在1978年的工作导致了abc猜想，2007年，他首称有了这个猜想的一个证明，但人们很快就发现它是有缺陷的。
Like Szpiro, and also like British mathematician Andrew Wiles, who proved Fermat’s Last Theorem in 1994, Mochizuki has attacked the problem using the theory of elliptic curves — the smooth curves generated by algebraic relationships of the sort y2=x3+ax+b.
像Szpiro，也像英国数学家安德鲁•怀尔斯---他在1994年证明了费尔马大定理---一样, 望月利用椭圆曲线的理论 ---由那种代数关系式y^2= x^3+ ax + b所产生的光滑曲线---挑战这个问题。
//// of the sort y2=x3+ax+b.应当是of the sort y^2=x^3+ax+b. 译者注///
There, however, the relationship of Mochizuki’s work to previous efforts stops. He has developed techniques that very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices. “At this point, he is probably the only one that knows it all,” says Goldfeld.
但是，望月停止了先前努力的相关工作。他推演了一些技术，真的没有几个数学家完全理解，这些技术引用了新的数学“物体”--- 抽象的实体，类似于我们比较熟悉的例子，如几何体，集合，排列，拓扑和矩阵。戈德菲尔德说“在这一点上，他可能是唯一一个知道这一切的家伙”。
Conrad says that the work “uses a huge number of insights that are going to take a long time to be digested by the community”. The proof is spread across four long papers1–4, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors,” Conrad explains.
康拉德说，这项工作“采用了数量庞大的见解，会需要很长的时间由这个社团消化”。证明分布在4个长篇的论文1-4卷里，每一篇都依赖于前面的长篇论文。 “这可能需要投入大量的时间去了解一个长而巧妙的证明，所以大家努力做这些后续工作不仅在于声明的重要性，但也在于作者的思路记录”康拉德继续解释道。
Mochizuki’s track record certainly makes the effort worthwhile. “He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence,” says Conrad. And he adds that the pay-off would be more than a matter of simply verifying the claim. “The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory.”
望月的历史记录的确是值得这个努力的。“过去，他已经证明了一些极为深刻的定理，在他的著作中，这是非常透彻的，所以它提供了大量的信心，”康拉德说。这个成果不只是简单地证明这一声称的问题。他又补充道，“令人兴奋的方面是，不只是猜想可能现在已经解决了，但他不得不引进的技术和见解也将是非常强大的工具，会用于解决未来的数论问题”。
This article is reproduced with permission from the magazine Nature. The article was first published on September 10, 2012.
本文转载自然“杂志的许可。文章首次发表于2012年9月10日。 |
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发表于 2012-10-27 22:06
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