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# 数学代写|代数数论代写Algebraic Number Theory代考|MA58400 Diophantine Equations

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## 数学代写|代数数论代写Algebraic Number Theory代考|Diophantine Equations

In this section, we look at the equation
$$x^2+k=y^3,$$

which was first introduced by Bachet in 1621 , and has played a fundamental role in the development of number theory. When $k=2$, the only integral solutions to this equation are given by $y=3$ (see Exercise 2.4.3); and this result is due to Fermat. It is known that the equation has no integral solution for many different values of $k$. There are various methods for discussing integral solutions of equation (6.2). We shall present, here, the one that uses applications of the quadratic field $\mathbb{Q}(\sqrt{-k})$, and the concept of ideal class group. This method is usually referred to as Minkowski’s method. We start with a simple case, when $k=5$.

Example 6.3.1 Show that the equation $x^2+5=y^3$ has no integral solution.

Solution. Observe that if $y$ is even, then $x$ is odd, and $x^2+5 \equiv 0(\bmod 4)$, and hence $x^2 \equiv 3(\bmod 4)$, which is a contradiction. Therefore, $y$ is odd. Also, if a prime $p \mid(x, y)$, then $p \mid 5$, so $p=5$; and hence, by dividing both sides of the equation by 5 , we end up with $1 \equiv 0(\bmod 5)$, which is absurd. Thus, $x$ and $y$ are coprime.

Suppose now that $(x, y)$ is an integral solution to the given equation. We consider the factorization
$$(x+\sqrt{-5})(x-\sqrt{-5})=y^3,$$
in the ring of integers $\mathbb{Z}[\sqrt{-5}]$.

## 数学代写|代数数论代写Algebraic Number Theory代考|Exponents of Ideal Class Groups

The study of class groups of quadratic fields is a fascinating one with many conjectures and few results. For instance, it was proved in 1966 by H. Stark and A. Baker (independently) that there are exactly nine imaginary quadratic fields of class number one. They are $\mathbb{Q}(\sqrt{-d})$ with $d=1,2,3,7,11,19,43,67,163$.

By combining Dirichlet’s class number formula (see Chapter 10, Exercise 10.5.12) with analytic results due to Siegel, one can show that the class number of $\mathbb{Q}(\sqrt{-\bar{d}})$ grows like $\sqrt{d}$. More precisely, if $h(-d)$ denotes the class number,
$$\log h(-d) \sim \frac{1}{2} \log d$$
as $d \rightarrow \infty$.
The study of the growth of class numbers of real quadratic fields is more complicated. For example, it is a classical conjecture of Gauss that there are infinitely many real quadratic fields of class number 1 . Related to the average behaviour of class numbers of real quadratic fields, C. Hooley formulated some interesting conjectures in 1984.

Around the same time, Cohen and Lenstra formulated general conjectures about the distribution of class groups of quadratic fields. A particular case of these conjectures is illustrated by the following. Let $p$ be prime $\neq 2$. They predict that the probability that $p$ divides the order of the class group of an imaginary quadratic field is
$$1-\prod_{i=1}^{\infty}\left(1-\frac{1}{p^i}\right)$$

## 数学代写|代数数论代写Algebraic Number Theory代考|Diophantine Equations

$$x^2+k=y^3,$$

$$(x+\sqrt{-5})(x-\sqrt{-5})=y^3,$$

## 数学代写|代数数论代写Algebraic Number Theory代考|Exponents of Ideal Class Groups

$$\log h(-d) \sim \frac{1}{2} \log d$$

$$1-\prod_{i=1}^{\infty}\left(1-\frac{1}{p^i}\right)$$

## MATLAB代写

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