Infinite descent

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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} = \frac{p}{q},

where p and q are relatively prime integers; in other words, the fraction is reduced to lowest terms. Then,

Failed to parse (Missing texvc executable; please see math/README to configure.): 2 = \frac{p^2}{q^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): 2q^2 = p^2, \,

so Failed to parse (Missing texvc executable; please see math/README to configure.): 2|p . Let Failed to parse (Missing texvc executable; please see math/README to configure.): p=2P , and

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle{2q^2 = (2P)^2 = 4P^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): q^2 = 2P^2, \,

so Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): a^2+b^2=3 \cdot (s^2+t^2),\,

then there will certainly be a minimal solution among them.

Suppose that Failed to parse (Missing texvc executable; please see math/README to configure.): a_1, b_1, s_1, t_1

is the minimal integer solution, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): a_1^2+b_1^2 = 3 \cdot (s_1^2+t_1^2)

and this is only true if both Failed to parse (Missing texvc executable; please see math/README to configure.): a_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): b_1
are divisible by 3. Set 

Failed to parse (Missing texvc executable; please see math/README to configure.): 3 a_2 = a_1\,

and Failed to parse (Missing texvc executable; please see math/README to configure.):  3 b_2 = b_1.\,

Thus we have

Failed to parse (Missing texvc executable; please see math/README to configure.): (3 a_2)^2 + (3 b_2)^2 = 3 \cdot (s_1^2+t_1^2)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

See also

• Proof by contradictionfr:Méthode de descente infinie

it:Discesa infinita ja:無限降下法