Tuesday 10 April 2018

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

I was going through a video by Numberphile where they were talking about the Catalan’s conjecture.
This is another conjecture which is simple to state; however extremely difficult to prove; just like Collatz conjecture, about which I already have a blog in place.
At the very outset of the blog; let me bring it to your knowledge that this blog is just for learning and understanding and contains “very little”-to-“null” original work on the subject. However, this blog will be exhaustive and will contain a lot of references that will help a newbie (like myself) get into the depths of the conjecture and fully understand concepts used in its proof.
Let us now define the conjecture;
This statement was conjectured by Eugene Catalan (1814–1894) and was sent to the editor of Journal fur die Reine und Angewandte Mathematik.
The conjecture is as follows,
and are two powers of natural numbers whose values are consecutive (i.e., 8 and 9); the conjecture is for a mathematical statement such as,

The only solution of is for , and , is , , , .
Catalan’s conjecture was proven true by Preda Mihăilescu in 2002; the proof involves the theory of cyclotomic fields and Galois Modules.
So, as you see now, the breadth of the subject blew up! On second were talking about squares and cubes and now we are talking of Cyclotomic fields and Galois Modules!
Nevertheless, I will try and cover each topic in brief and quench our mathematical thirst!
Let us consider (with and ; for the sake of eliminating any confusion between and an english “a”), unless otherwise stated, and can be negative integers as well. Now, we re-write as follows,

The GCD of the two factors on the left hand side of the equation (after considering ) is either or (How did this happen?).

Some concepts before we move ahead:
The Wieferich pairs
In mathematics,
a Wieferich pair is a pair of prime number and that satisfy,

Let us write as, for,
This suggests a traditional approach of factorizing the left hand side
in , the ring of integers in the th cyclotomic
field

Ring:
A ring in the mathematical sense is a set together with two binary
operators and (Addition and multiplication), satisfying the
following conditions:
1. Additive associativity: For all , ,
2. Additive commutativity: For all , ,
3. Additive identity: There exists an element in such that for all a in , .
4. Additive inverse: For every a in there exists such that ,
5. Left and right distributivity: For all , and ,
6. Multiplicative associativity: For all , ( ring satisfying this property is sometimes
explicitly termed an associative ring). Conditions 1-5 are always
required. Though non-associative rings exist, virtually all texts also
require condition 6.
7. Multiplicative commutativity: For all , ( ring satisfying this property is termed a commutative ring),
8. Multiplicative identity: There exists an element such that for all , (a ring satisfying this
property is termed a unit ring, or sometimes a “ring with identity”),
9. Multiplicative inverse: For each , there exists an element such that , where is the identity element.

A brief history of the past developments on this conjecture is a must to be read and understood; the significance comes due to the period of 150 years for which it remained an open problem,
  • Only after six years after Catalan formally defined the conjecture, a result was proposed by French mathematician, Victor Lebesgue. He stated that, for the equation, ; where is a prime; has no solutions for positive values of and . A proof of the same will be discussed in brief in the later part of the blog.
  • After Lebesgue’s work, all development solely consisted of small exponents, and then Naggel showed in 1921 that the difference between a third power and an other perfect power never is equal to 1.
  • In 1932, Selberg proved that, has no solution in positive integers when . A stronger result to this was proved by Ko Cho in 1965, that stated that the equation has no solutions for positive integers when .
  • Cassels made some observations for where and are odd-primes. He proved that is this equality holds for positive integers and , then divides and divides . For the case , this had already been shown by Naggel
  • Inkeri defined the concept of a Wieferich pair [the definition and explanation of the same is given above] in the concept of Catalan equation as follows:
    If the Catalan’s equation holds, then either is a Wieferich Pair, or divides , the class number of the cyclotomic field , or divides , the class number of the cyclotomic field

Cyclotomic Field:
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to , the field of rational numbers. The -th cyclotomic field (where ) is obtained by adjoining a primitive -th root of
Primitive root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives when raised to some positive integer power .

  • Some time later Mihailescu proved that the Catalan equation has no solutions if and are odd and does not divide . By this result the Catalan conjecture became a theorem


More on Cyclotomic Fields:
Let be an odd-prime number. Let be the -th cyclotomic polynomial in i.e., . Consider the field extension of , where denotes a primitive -th root of unity. This is a field extension of degree and it is reducible in . We denote by from now on.
This field extension is Galois with Galois Group,

Since the map,


is an isomorphism
The automorphism acts in all embeddings as complex conjugation. Therefore, we call complex conjugation.
The fixed field of complex conjugation is , which is called the maximal real subfield of . We denote by . The field extension of has degree and it is Galois with Galois theory.
Another important concept that is

Mihailescu’s proof

To be contd…in the next blogpost!

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