$(x, y)\oplus (z, w) = (xz − wy, xw + yz).$

1.  没有有理点.

$x^2+y^2=3$

2. 恰有一个有理点.

3. 恰有两个有理点.

$x^2+(y+\sqrt2)^2=3$

4. 有无穷个有理点  这是我们关心的情形.

\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

$x^2+y^2=a^2+b^2,$

$(x+a)(x-a)=(b+y)(b-y).$

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

\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}

\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}

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

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

$2t(a+bu)+t^2(u^2+1)=0$

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

\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}

$ax^2+bxy+cy^2+dx+ey+f=0$

## 中心非有理点

$l\pi+me+n=0.$

Counting from Infinity, A film about Yitang Zhang and the twin prime conjecture, 终于公映了

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 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”.

ICM 2014 Program

4 个数学家将获得本届大会的 Fields Medal.

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

2. 关于 Siegel 零点
2014.7.2.9:30-11:30

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.

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

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

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

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

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

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