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. For other meanings of mathematics or math, see Mathematics (disambiguation) and Math (disambiguation). ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...

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. In mathematics, the method of considering a minimal counterexample combines the ideas of inductive proof and proof by 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). The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ... In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... This article or section does not cite its references or sources. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... Louis Joel Mordell (28 January 1888 - 12 March 1972) was a British mathematician, known for pioneering research in number theory. ... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

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. In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. ... In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ... In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. ...

Simple application examples

Irrationality of √2

Suppose that √2 were rational. Then it could be written as In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

where p and q are relatively prime integers; in other words, the fraction is reduced to lowest terms. Then, In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

so 2 | p. Let p=2P, and

so 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

A Diophantine equation

Suppose there are integer solutions of

then there will certainly be a minimal solution among them.

Suppose that a₁,b₁,s₁,t₁ is the minimal integer solution, we have

and this is only true if both a₁ and b₁ are divisible by 3. Set

and

Thus we have

and

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. In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ...

See also

• Mathematical induction
• Proof by contradiction