Catalan's Conjecture - A learning exercise for the bored mind - Contd.

Before we move to Mihailescu’s proof, let us cover some more mathematical concepts (so that we feel superior to the people around us, just kidding :smile:),

There is an interesting theorem that Mihailescu uses in his proof, called the Stickelberger’s theorem,

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Stickelberger’s theorem:

This is a result of algebraic number theory, which gives more information about Galois Module structure of class groups of Cyclotomic Fields.
This theorem consists of Stickelberger’s element and Stickelberger’s ideal.
I will now state the complete definition of the theorem and then visit its corners as we move along,

Let $K_m$ denote the $m$-th cyclotomic field${[1]}$ . It is a Galois extension of $\mathbb{Q}$ with Galois Group $G_m$ isomorphic to the multiplicative group of integers modulo $m$ $(\mathbb{Z}/m\mathbb{Z})^{\times}$.

The Stickelberger element (of level $m$ or of $K_m$) is an element in thr group ring $\mathbb{Q}[G_m]$ and the Stickelberger ideal is an ideal in the group ring $\mathbb{Z}[G_m]$. (Note: $\mathbb{Z} \subset \mathbb{Q}$).

The definition of both the Stickelberger element and ideal are, let $\zeta_m$ denote a primitive $m$-th root of unity ${[2]}$. The isomorphism from $(\mathbb{Z/mZ})^{\times}$ to $G_m$ is given by sending $a$ to $\sigma_a$ by the relation $$\sigma_a(\zeta_m)=\zeta^{a}_m.$$
The Stickelberger element of level $m$ is given by,
$$\theta(K_m)=\frac{1}{m}\sum_{a=1 \ (a,m)=1}^ma\cdot \sigma_a^{-1} \in \mathbb{Q}[G_m].$$
The Stickelberger ideal of level $m$ is given by,
$$I(K_m)=\theta(K_m) \mathbb{Z}[G_m] \cap \mathbb{Z}[G_m].$$

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Inkeri$[3]$, used the concept of Wiefrich pair (explain in the previous blogpost of this series) in the context of Catalan’s equation as follows:

A Wieferich pair is a pair $(p,q)$ of primes such that $$p^{q-1} \equiv 1( \text{mod} \ q^2)$$ and $$q^{p-1} \equiv 1(\text{mod} \ p^2).$$

He showed that if the Catalan’s equation holds, then either $(p,q)$ is a Wieferich pair, or $q$ divides $h_p$, the class number$[4]$ of cyclotomic field $\mathbb{Q}(\zeta_p)$, or $p$ divides $h_q$, the class number of cyclotomic field $\mathbb{Q}(\zeta_q)$, there were other developments in this direction too.

Bugeaud and Hanrot $[5]$proved a class number criterion concerning Catalan’s equation, which implies that the Catalan’s Equation ( $x^p - y^q =1$) has no solutions in non-zero integers $x$ and $y$ if $p$ and $q$ are primes such that one of them is smaller than 43. This was a huge achievement , I recommend you to have a look at the paper at [5].

Mihailescu proved that the Catalan equation has no solutions if $p$ and $q$ are odd and $q$ does not divide $p - 1$. By this result the Catalan conjecture became a theorem. And later Mihailescu succeeded in finding a more elegant proof of Catalan’s conjecture in the case where $q$ does divide $p-1$. Thus, Catalan’s conjecture is a theorem with an algebraic proof in which no computer calculations
are needed.

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In this section e wll discuss some breakthrough results by Mihailescu. The most important one is that $q^2$ divides $x$.

The following lemma will be used for that, an element $a$ of a ring $R$ is called nilpotent if $\exists$ and integer $n$ such that $a^n=0$.

Lemma 1: The ring $\mathbb{O}_k/(q)$ does not contain nilpotent elements, if $\alpha$, $\beta$ $\in$ $\mathbb{O}_k = \mathbb{Z}[\zeta]$ satisfy the congruence $\alpha^p \equiv \beta^q (\text{mod} \ q^2)$

Theorem 1: For $\theta \in {I_s}^{-}$, the element $(x-\zeta)^\theta$ is a $q^{th}$ in $K=\mathbb{Q}(\zeta)$. We also have that $q^2$ divides $x$ and $p^2$ divides $y$.

The proofs of the above Lemma and Theorem are beyond the scope of this blog; regardless to say that the pre-requisites are already covered in detail. For the interested, you can refer Catalan’s Conjecture - A cyclotomic field.

Cheers!