Infinite descent question ?

Question

i don't understand the principles behind infinite descent and how to apply them.

can someone give me an example of how to use infinite descent?

also does infinite descent prove that an equation has No integer solutions OR Infinite number of integer solutions

thanks in advance

Answer 1

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 is 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 x4 + y4 = z2 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).