In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction[1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction.
[3][4] Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers.
However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction.
[2] The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations.
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 some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doubling function for rational points on an elliptic curve E. The context is of a hypothetical non-trivial rational point on E. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits), so that "halving" a point gives a rational with smaller terms.
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.
In this way, abstractly-defined cohomology groups in the theory become identified with descents in the tradition of Fermat.
The proof that the square root of 2 (√2) is irrational (i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the ancient Greeks, and is perhaps the earliest known example of a proof by infinite descent.
Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned.
For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.
[9] The ancient Greeks, not having algebra, worked out a geometric proof by infinite descent (John Horton Conway presented another geometric proof by infinite descent that may be more accessible[10]).
[11] (Alternatively, this proves that if √2 were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation p/q would imply that a smaller one existed, which is a similar contradiction.)
Then The numerator and denominator were each multiplied by the expression (√k − q)—which is positive but less than 1—and then simplified independently.
in integers, which is a special case of Fermat's Last Theorem, and the historical proofs of the latter proceeded by more broadly proving the former using infinite descent.
Then it can be scaled down to give a primitive (i.e., with no common factors other than 1) Pythagorean triangle with the same property.
; since such a sequence cannot go on infinitely, the original premise that such a triangle exists must be wrong.
For other similar proofs by infinite descent for the n = 4 case of Fermat's Theorem, see the articles by Grant and Perella[14] and Barbara.