$\{\sqrt n|n\in\Bbb N\; \text{is Square-free integer}\}$

wiki 给出的定义是: 不被不是 $$1$$ 的完全平方整除的整数称为无平方因子整数. 因此, $$1$$ 算无平方因子正整数. 鉴于此, 我们认为: 无平方因子整数定义为”不被质数的平方整除的整数”更为恰当.

$$n_1$$, $$n_2$$, $$\dotsc$$, $$n_k$$ 是互不相同的无平方因子正整数; $$a_1$$, $$a_2$$, $$\dotsc$$, $$a_k$$ 都是整数. 令

$S=a_1\sqrt{n_1}+a_2\sqrt{n_2}+\dotsb+a_k\sqrt{n_k},$

$S^\prime=b_1\sqrt{m_1}+b_2\sqrt{m_2}+\dotsb+b_l\sqrt{m_l},$

(这里 $$b_1$$, $$b_2$$, $$\dotsc$$, $$b_l$$ 都是整数; $$m_1$$, $$m_2$$, $$\dotsc$$, $$m_l$$ 都是无平方因子正整数, 并且都没有 $$p_1$$, $$p_2$$, $$\dotsc$$, $$p_N$$ 以外的质因子), 使得 $$SS^\prime\ne0$$ 是整数. 进而, 顺水推舟, $$S\ne0$$.

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

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.

$$x^2+y^2=\frac pq$$

($$p, q$$ 是互质的正整数) 即可.

## 中心是有理点

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

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

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

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

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

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

$$x=a+t, \; y=b+tu.$$

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

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

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

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

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

## 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

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 1

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 2

Yitang Zhang is giving the last invited talk 3

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.

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

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