Jul 042015
 

圆周上的有理点有这么几个情况.

1.  没有有理点.

\[x^2+y^2=3\]

是一个例子.

2. 恰有一个有理点.

比如 \((x+\sqrt2)^2+(y+\sqrt2)^2=4\) 只有原点.

3. 恰有两个有理点.

比如

\[x^2+(y+\sqrt2)^2=3\]

4. 有无穷个有理点  这是我们关心的情形.

很容易证明, 如果一个圆周上有 \(3\) 个有理点, 则有无穷多个有理点在此圆周上, 并且此圆的圆心是有理点, 半径的平方是有理数. 所以, 只要关注

\begin{equation}x^2+y^2=\frac pq\end{equation}

(\(p, q\) 是互质的正整数) 即可.

中心是有理点

方程 \((1)\) 有有理解当且仅当 \(p, q\) 都能表成两个整数的平方和. 在有有理解的情形下, 有无穷多个有理点在这个圆周上, 也就是有无穷多对有理数的平方和是 \(\dfrac pq\).

有几个办法可以定出 \((1)\) 的有理解.

初等一点的办法, 可以从

\[x^2+y^2=a^2+b^2, \]

着手, 这里 \((a,b)\) 是 \((1)\) 的一组有理解, 即  \(a^2+b^2=\dfrac pq\) 为真. 于是

\[(x+a)(x-a)=(b+y)(b-y). \]

当 \(y+b\ne0\), 且 \(x-a\ne0\) 时,

\[\frac{x+a}{y+b}=\frac{b-y}{x-a}. \]

设这个值是有理数 \(t\). 由

\begin{equation}\frac{x+a}{y+b}=\frac{b-y}{x-a}=t\end{equation}

\begin{cases}x+a=t(y+b)\\ t(x-a)=b-y.\end{cases}

这是很简单的线性方程组. 根据 Cramer 法则

\begin{equation}\begin{cases}x=\frac{(t^2-1)a+2bt}{t^2+1}\\ y=\frac{(1-t^2)b+2at}{t^2+1}.\end{cases}\end{equation}

但这并不是我们要寻找的全部有理解, 但也仅仅只有一个点 \((a,-b)\) 不在其中.

下面是一种普遍有效的办法, 这也是代数几何, 椭圆曲线的理论所采纳的途径, 尽管表现形式可能有别.

\begin{equation}x=a+t, \; y=b+tu.\end{equation}

带入 \((1)\) 得

\[(a+t)^2+(b+tu)^2=\frac pq.\]

注意 \(a^2+b^2=\dfrac pq\), 因此

\[2t(a+bu)+t^2(u^2+1)=0\]

导出 \(t=0\) 或

\begin{equation}t=-\frac{2(a+bu)}{u^2+1}.\end{equation}

记得 \((4)\), 于是

\begin{equation}x=a-\frac{2(a+bu)}{u^2+1}=\frac{(u^2-1)a-2bu}{u^2+1}, \; y=b-\frac{2u(a+bu)}{u^2+1}=\frac{(1-u^2)b-2au}{u^2+1}.\end{equation}

从 \((2)\), \((4)\) 看出: \((3)\) 中的 \(t\) 与 \((6)\) 中的 \(u\) 都与两点 \((x,y)\)  与 \((a,b)\) 所成直线的斜率有关, 只不过 \(t\) 表示这个斜率的相反数, 而 \(u\) 就是这个斜率. 这也轻松的解释了为什么 \((3)\) 与 \((6)\) 给出了圆上除 \((a,-b)\) 这一个点以外所有的有理点.

这个手段可以用来搜索更一般的二次曲线

\[ax^2+bxy+cy^2+dx+ey+f=0\]

上的有理点.

中心非有理点

圆心不是有理点的圆上有简洁明了的一般性的结论么? 遗憾的是, 没有这样利索的定理. 随便找一个(中心非有理点的)圆, 很可能(暂时)无法弄清楚到底有几个有理点.

举个例子, 中心在 \((\pi,e)\) 的圆上有几个有理点?

看起来没啥特别, 但这只是表象, 这其实是一个完全不同的问题: \(1,\pi,e\) 在有理数域上线性无关, 即是否存在不全为 \(0\) 的有理数 \(l\), \(m\), \(n\), 使得

\[l\pi+me+n=0.\]

这是一个很有名的 open problem.

我们相信中心在 \((\pi,e)\) 的圆上有至多 \(1\) 个有理点. 遗憾的是, 暂时无法提供理由来说明不能有 \(2\) 个.

 Posted by at 9:05 am
Aug 212014
 

Yitang Zhang is giving the last invited talk at ICM 2014, “Small gaps between primes and primes in arithmetic progressions to large moduli”.

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 1

这是闭幕式前的最后一个 invited talk. 张大师习惯手写, 当场演算.

Yitang Zhang stepped onto the main stage of mathematics last year with the announced of his achievement that is hailed as “a landmark  theorem in the distribution of prime numbers”.

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 2

Yitang Zhang is giving the last invited talk

Yitang Zhang is giving the last invited talk 3

Aug 052014
 

ICM 2014 Program
这届国际数学界大会(International Congress of Mathematicians, ICM)的安排, 已经明确无误的说明:

 4 个数学家将获得本届大会的 Fields Medal.

张大师将于 8 月 21 日作 ICM 闭幕式之前的压轴报告, 这是只有今年的 Fields Medalist, Gauss Prize, Chern Medal 得主才有的殊荣

张益唐 7 月1 日在北大本科生毕业典礼有一个讲话

这个暑假, 张大师在中国科学院晨兴数学中心和他的母校北京大学做了好几次讲座.

1. A Transition Formula for Mean Values of Dirichlet Polynomials
2014,6.23./6.25. 9:30-11:30
晨兴 110
主持人: 王元

2. 关于 Siegel 零点
2014.7.2.9:30-11:30
晨兴 110

3. Distribution of Prime Numbers and the Riemann Zeta Function
July 8, 10, 2014 16:00-17:00, 镜春园82号甲乙丙楼的中心报告厅
July 15, 16:30-17:30 镜春园 78 号院的 77201 室.
主持人: 刘若川

4. 关于 Siegel 零点(2)
014.7.16./7.30./8.4./8.6. 9:30-11:30
N820

Jul 292014
 

I’ve just received a book named Number Theory in the Spirit of Liouville by Kenneth S. Williams.

Number Theory in the Spirit of Liouville

Number Theory in the Spirit of Liouville

Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville’s ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.

  • Demonstrates that some analytic formulae in number theory can be proved in an elementary arithmetic manner
  • Motivates students to do their own research
  • Includes an extensive bibliography

Table of Contents

Preface
1. Joseph Liouville (1809–1888)
2. Liouville’s ideas in number theory
3. The arithmetic functions \(\sigma_k(n)\), \(\sigma_k^*(n)\), \(d_{k,m}(n)\) and \(F_k(n)\)
4. The equation \(i^2+jk = n\)
5. An identity of Liouville
6. A recurrence relation for \(\sigma^*(n)\)
7. The Girard–Fermat theorem
8. A second identity of Liouville
9. Sums of two, four and six squares
10. A third identity of Liouville
11. Jacobi’s four squares formula
12. Besge’s formula
13. An identity of Huard, Ou, Spearman and Williams
14. Four elementary arithmetic formulae
15. Some twisted convolution sums
16. Sums of two, four, six and eight triangular numbers
17. Sums of integers of the form \(x^2+xy+y^2\)
18. Representations by \(x^2+y^2+z^2+2t^2\), \(x^2+y^2+2z^2+2t^2\) and \(x^2+2y^2+2z^2+2t^2\)
19. Sums of eight and twelve squares
20. Concluding remarks
References
Index.

Review

“… a fascinating exploration and reexamination of both Liouville’s identities and “elementary” methods, providing revealing connections to modern techniques and proofs. Overall, the work contributes significantly to both number theory and the history of mathematics.”

J. Johnson, Choice Magazine

Publisher: Cambridge University Press (November 29, 2010)
Language: English
FORMAT: Paperback
ISBN: 9780521175623
LENGTH: 306 pages
DIMENSIONS: 227 x 151 x 16 mm
CONTAINS: 275 exercises

Jul 272014
 

I’ve just received a book named Development of Elliptic Functions According to Ramanujan
by K Venkatachaliengar (deceased) , edited by: ShaunCooper , Shaun Cooper

Development of Elliptic Functions According to Ramanujan

Development of Elliptic Functions According to Ramanujan

This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter.

The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan’s work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

Contents:

  • The Basic Identity
  • The Differential Equations of \(P\), \(Q\) and \(R\)
  • The Jordan-Kronecker Function
  • The Weierstrassian Invariants
  • The Weierstrassian Invariants, II
  • Development of Elliptic Functions
  • The Modular Function \(\lambda\)

Readership: Graduate students and researchers in Number Theory and Classical Analysis, as well as scholars and aficionados of Ramanujan’s work.

Review

It is obvious that every arithmetician should want to own a copy of this book, and every modular former should put it on his ‘to be handled-with-loving-care-shelf.’ Reader of Venkatachaliengar’s fine, fine book should be willing to enter into that part of the mathematical world where Euler, Jacobi, and Ramanujan live: beautiful formulas everywhere, innumerable computations with infinite series, and striking manouevres with infinite products.

— MAA Reviews

The author was acquainted with many who knew Ramanujan, and so historical passages offer information not found in standard biographical sources. The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers.

— MathSciNet

The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers. italic Zentralblatt MATH

— Zentralblatt MATH

  • Series: Monographs in Number Theory (Book 6)
  • Hardcover: 184 pages
  • Publisher: World Scientific Publishing Company (September 28, 2011)
  • Language: English
  • ISBN-10: 9814366455
  • ISBN-13: 978-9814366458
Jul 242014
 

Conjecture

There exist elliptic curve groups \(E(\Bbb Q)\) of arbitrarily large rank.

用 \(r\) 表示 \(\Bbb Q\) 上的椭圆曲线 \(E\) 的秩—the rank of the Mordell–Weil group \(E(\Bbb Q)\).

一个悬而未决的著名难题是: \(r\) 是否可以任意大?

Martin-McMillen 2000 年有一个 \(r\geq24\) 的例子:

\begin{equation*}\begin{split}y^2+xy+y&=x^3-120039822036992245303534619191166796374x\\&+ 504224992484910670010801799168082726759443756222911415116\end{split}\end{equation*}

Hasse-Weil \(L\)-function \(L(s, E)\) 在 \(s=1\) 处的零点的阶数 \(r_a\) 称为 \(E\) 的 analytic rank(解析秩).

Manjul Bhargava, Christopher Skinner, Wei Zhang(张伟) 7 月 7 日在 arXiv 上传的论文 “A majority of elliptic curves over \(Q\) satisfy the Birch and Swinnerton-Dyer conjecture“, 宣布了取得的进展:

  1. \(\Bbb Q\) 上的椭圆曲线, when ordered by height(同构类以高排序), 至少有 \(66.48\%\) 满足 BSD conjecture;
  2. \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(66.48\%\) 有有限 Tate–Shafarevich group;
  3. \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(16.50\%\) 满足 \(r=r_a=0\), 至少有 \(20.68\%\) 满足 \(r=r_a=1\).

谁将在 8 月 13 日的 ICM 2014 开幕式上获得 Fields medal?坊间向来不缺传闻. 数论大牛 Manjul Bhargava 无疑是最耀眼的明星.