数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

Number Theory is the study of numbers, in particular the whole numbers $1,2,3, \ldots$, also called the natural numbers. The set of natural numbers is denoted by $\mathbb{N}$. Leaving aside the unit 1 , these numbers fall into two categories: The indivisible numbers $2,3,5,7, \ldots$ are the primes, and the rest $4,6,8,9,10, \ldots$ composed of primes, are the composite numbers. The following basic facts, with proofs, about these numbers were already known to Euclid around 300 B.C.
Theorem 1.1. There are infinitely many primes.
Theorem 1.2 (Fundamental Theorem of Arithmetic). Every natural number $n>1$ is a unique product
$$n=p_{1}^{e_{1}} \ldots p_{r}^{c_{r}} \quad(r \geq 1)$$
of powers of distinct primes $p_{1}, \ldots, p_{r}$, taken in some order.
By looking at the list of primes, one can ask several naive but still unanswered questions. For example, is there an endless supply of twin primes? We call a pair of primes $q, p$ twin primes if $p=q+2$. [This is the closest two odd primes can be to each other.] A glance at the list
$$3,5 ; 5,7 ; 11,13 ; 17,19 ; 29,31 ; \ldots$$
suggests that there are infinitely many pairs of twin primes, but no one has ever been able to prove this so far. Another big problem in number theory is the unproven conjecture of Goldbach, which asserts that every even number larger than 2 is a sum of two primes.

Many questions in number theory arise naturally in the study of geometry. The most fundamental fact in Euclidean geometry is the theorem of Pythagoras, which may be called the fundamental theorem of geometry. Actually, it was known to the Egyptians and Babylonians about two thousand years earlier, but they had no rigorous proof of it like Euclid did.

数学代写|代数数论代写Algebraic number theory代考|Methods of Proving Theorems in Number Theory

The method that has been used since antiquity is the unique factorization. Let us recall Euclid’s proof of Theorem 1.1.

It follows from the unique factorization (1.1) that any $n>1$ is either a prime or has a prime factor. To prove Theorem $1.1$ by contradiction, suppose there are only finitely many primes, say $p_{1}, \ldots, p_{r}$. Now consider the number $n=p_{1} \ldots p_{\mathrm{r}}+1$. It is not a prime because it is larger than every prime $p_{j}$. So, it has a prime factor, say $p_{1}$. Therefore $n=p_{1} a$ for an integer $a$. This implies that $1=p\left(a-p_{2} \ldots p_{r}\right)$. This is a contradiction because 1 has no prime factor.

Another example of such a proof is the proof below by Euler (1770) of the following claim of Fermat (1657): 27 is the only cube that exceeds a square by 2. In modern terminology, $(3, \pm 5)$ are the only points with integer coordinates on the elliptic curve
$$y^{2}=x^{3}-2 .$$
Proof. In the ring $\mathbb{Z}[\sqrt{-2}]={a+b \sqrt{-2} \mid a, b \in \mathbb{Z}}$, which is a UFD (see Exercise 8, Chapter 2), we use the factorization
$$x^{3}=y^{2}+2=(y+\sqrt{-2})(y-\sqrt{-2}) .$$
In general, in a UFD, if $\alpha, \beta$ have no common factor other than units, and $\alpha \beta=\gamma^{m}$ for an integer $m>0$, then $\alpha=\alpha_{1}^{m}$ and $\beta=\beta_{1}^{m}$ for some $\alpha_{1}, \beta_{1}$ in it. Therefore
$$y+\sqrt{-2}=(a+b \sqrt{-2})^{3} \text { for } a, b \in \mathbb{Z} .$$
By expanding $(a+b \sqrt{-2})^{3}$ and comparing the real/imaginary parts, we get
$$1=b\left(3 a^{2}-2 b^{2}\right), y=a^{3}-6 a b^{2} .$$
But the first equation in (1.7) can hold only if $b=1$ and $a=\pm 1$. This implies $y=\pm 5$.

1. Analytic Methods
Euler initiated what we call the analytic number theory. The study of infinite series (analysis) can lead to interesting results in number theory. Let us recall Euler’s proof of the infinitude of primes. Leaving aside the issue of convergence, by multiplying the infinite series formally, one sees that
\begin{aligned} \sum_{n=1}^{\infty} \frac{1}{n}=& \sum_{n=1} \frac{1}{p_{1}^{e_{1}} \ldots p_{r}^{e_{r}}}=\prod_{p}\left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right), \text { i.e. } \ & \frac{1}{n}=\prod_{p}^{\infty}\left(1-\frac{1}{p}\right)^{-1} \end{aligned}
the product (called the Euler product) taken over all primes $p$. Note that the first equality is a consequence of the unique factorization (1.1).

The partial sums $\sum_{n=1}^{N-1} \frac{1}{n}$ of the series $\sum_{n=1}^{\infty} \frac{1}{n}$ are bounded from below by the area (cf. Figure 1.1) $\int_{1}^{N} \frac{d x}{x}=\ln N$, which goes to infinity as $N$ goes to infinity.

数学代写|代数数论代写Algebraic number theory代考|Techniques from Algebraic Geometry

Algebraic geometry is the study of the solutions of polynomial equations in a number of variables $x_{1}, \ldots, x_{n}$ with values of $x_{j}$ in a field $K$. Unless we assume $K$ to be algebraically closed, such as the field $\mathbb{C}$ of complex numbers, the subject is not satisfactory. For example, $x^{2}+y^{2}+1=0$ has no solution with $x, y$ even in such a big field as $\mathbb{R}$, the field of real numbers. Moreover, a line (equation of degree 1) is supposed to meet a circle (equation of degree 2) in two points. This rarely happens, but happens every time (in the projective plane $\mathbb{P}^{2}(\mathbb{C})$ ), thanks to Bezout’s Theorem: Two curves of degree $d_{1}, d_{2}$ with no component in common intersect in $d_{1} d_{2}$ points in the projective plane $\mathbb{P}^{2}(\mathbb{C})$, counted properly.

The arithmetic algebraic geometry is the subject in which algebraic geometric methods are used to answer questions in number theory. We illustrate it by finding the primitive Pythagorean triples, which is the same as finding the rational points (points with rational coordinates) on the unit circle
$$X^{2}+Y^{2}=1$$
with the rational numbers $X, Y$ in the lowest form. A primitive Pythagorean triple $(x, y, z)$ gives such a rational point with $X=\frac{x}{z}, Y=\frac{y}{z}$, and vice versa.

To obtain an algorithm to find all the primitive Pythagorean triples $(x, y, z)$, we parameterize the unit circle (1.9) by the slope $t$ of the line through the fixed point $(-1,0)$ and a variable point $(X, Y)$ on this circle (cf. Figure 1.2).
Substituting for $X$ from the equation $X=t Y-1$ of this line in equation (1.9) of the unit circle, an easy calculation shows that
$$Y=\frac{2 t}{1+t^{2}} \text { and } X=t Y-1=\frac{1-t^{2}}{1+t^{2}} .$$
If we run $t$ through all rational numbers in the lowest form $t=\frac{a}{b}$, we get the following result:
FIGURE 1.2: Rational points on the unit circle.
Theorem 1.6. Every primitive Pythagorean triplet $(x, y, z)$ is of the form
$$x=a^{2}-b^{2}, y=2 a b, z=a^{2}+b^{2},$$
where $a, b(a>b)$ are positive integers of opposite parity (one odd, the other even) with no common factor.

Note that the condition of opposite parity is necessary because otherwise $x$, $y, z$ are all even, so $(x, y, z)$ is not primitive. We also remark that switching $x$ and $y$ does not produce a different Pythagorean triplet.

数学代写|代数数论代写Algebraic number theory代考|What Is Number Theory

。定理 1.1。有无穷多个素数。

n=p1和1…prCr(r≥1)

3,5;5,7;11,13;17,19;29,31;…

数学代写|代数数论代写Algebraic number theory代考|Methods of Proving Theorems in Number Theory

X3=是2+2=(是+−2)(是−−2).

1=b(3一个2−2b2),是=一个3−6一个b2.

1. 解析方法
欧拉开创了我们所说的解析数论。对无穷级数（分析）的研究可以在数论中产生有趣的结果。让我们回顾一下欧拉关于素数无穷大的证明。撇开收敛问题不谈，通过形式上的无限级数相乘，可以看到
∑n=1∞1n=∑n=11p1和1…pr和r=∏p(1+1p+1p2+⋯), IE  1n=∏p∞(1−1p)−1
取所有素数的乘积（称为欧拉乘积）p. 请注意，第一个等式是唯一因式分解 (1.1) 的结果。

数学代写|代数数论代写Algebraic number theory代考|Techniques from Algebraic Geometry

X2+是2=1

X=一个2−b2,是=2一个b,和=一个2+b2,

有限元方法代写

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