Infinite descent - Citizendia

In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that 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. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield

This illustrative description can be restated in terms of a minimal counterexample, giving a more common type of formulation of an induction proof. In Mathematics, the method of considering a minimal counterexample (or minimal criminal) combines the ideas of Inductive proof and 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. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 It was developed by and much used for Diophantine equations by Fermat. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the 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. This article refers to the REM live recording For the mathematical term see Perfect square. In some cases, to a modern eye, what he was using was (in effect) the doubling mapping on an elliptic curve. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O 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). This article refers to the REM live recording For the mathematical term see Perfect square. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members

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. 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 The twentieth century of the Common Era began on In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory. 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. 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 x 1

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. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions 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 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 Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to 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, the Mordell–Weil theorem states that for an Abelian variety A over a Number field K, the group A (

1 Application examples
- 1.1 Irrationality of √2
- 1.2 A Diophantine equation
2 See also

Application examples

Irrationality of √2

Suppose that √2 were rational. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Then it could be written as

$\sqrt{2} = \frac{p}{q},$

where p and q are relatively prime integers; in other words, the fraction is reduced to lowest terms. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than Then,

$2 = \frac{p^2}{q^2}$

$2q^2 = p^2, \,$

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

$\displaystyle{2q^2 = (2P)^2 = 4P^2}$

$q^2 = 2P^2, \,$

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 in several different ways

A Diophantine equation

Suppose there are integer solutions of

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

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

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

and this is only true if both $a 1$ and $b 1$ are divisible by 3. Set

$3 a_2 = a_1\,$ and $3 b_2 = b_1.\,$

Thus we have

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

and

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

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. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only

Contents

Application examples

Irrationality of √2

A Diophantine equation

See also