Infinite descent Totally Explained

Everything about Infinite Descent totally explained

In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that's in some sense 'smaller'. Then one must show, usually with greater ease, that the infinite descent implied by having a whole sequence of solutions that are ever smaller, by our chosen measure, is an impossibility. This is a contradiction, so no such initial solution can exist. This illustrative description can be restated in terms of a minimal counterexample, giving a more common type of formulation of an induction proof. We suppose a 'smallest' solution - then derive a smaller one. That again is a contradiction.
The method can be seen at work in one of the proofs of the irrationality of the square root of two. It was developed by and much used for Diophantine equations by Fermat. Two typical examples are solving the diophantine equation

x^4+y^4=z^2

and proving a prime p ≡ 1 (mod 4) can be expressed as a sum of two perfect squares. In some cases, to a modern eye, what he was using was (in effect) the doubling mapping on an elliptic curve. More precisely, his method of infinite descent was an exploitation in particular of the possibility of halving rational points on an elliptic curve E by inversion of the doubling formulae. The context is of a hypothetical rational point on E with large co-ordinates. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits): so that a 'halved' point is quite clearly smaller. In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression).
In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory and the study of L-functions. The structural result of Mordell, that the rational points on an elliptic curve E form a finitely-generated abelian group, used an infinite descent argument based on E/2E in Fermat's style.
To extend this to the case of an abelian variety A, André Weil had to make more explicit the way of quantifying the size of a solution, by means of a height function - a concept that became foundational. To show that A(Q)/2A(Q) is finite, which is certainly a necessary condition for the finite generation of the group A(Q) of rational points of A, one must do calculations in what later was recognised as Galois cohomology. In this way, abstractly-defined cohomology groups in the theory become identified with descents in the tradition of Fermat. The Mordell-Weil theorem was at the start of what later became a very extensive theory.

Application examples

Irrationality of √2

Suppose that √2 were rational. Then it could be written as ť

sqrt

q^2 = 2P^2, ,

2|q

. But then 2 is a factor of both p and q, contradicting the fact that p and q are relatively prime. Since √2 is a real number, which can be either rational or irrational, the only option left is for √2 to be irrational.

A Diophantine equation

Suppose there are integer solutions of ť

a^2+b^2=3 cdot (s^2+t^2),,

then there will certainly be a minimal solution among them.
Suppose that

a_1, b_1, s_1, t_1

is the minimal integer solution, we have ť

a_1^2+b_1^2 = 3 cdot (s_1^2+t_1^2)

and this is only true if both

a_1

and

b_1

are divisible by 3. Set ť

3 a_2 = a_1,

and

3 b_2 = b_1.,

Thus we've ť

(3 a_2)^2 + (3 b_2)^2 = 3 cdot (s_1^2+t_1^2)

and ť

3(a_2^2+b_2^2) = s_1^2+t_1^2.,

which is a smaller solution — a contradiction, as the solution was assumed to be minimal! This shows that there are no nonzero solutions for this Diophantine equation.

Further Information

Get more info on 'Infinite Descent'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://infinite_descent.totallyexplained.com">Infinite descent Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.