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In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. One typical application is to show that a given equation has no solutions. 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. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction (repeating the same step) the original premise—that any solution exists—must be incorrect. It is disproven because its logical outcome would require a contradiction. An alternative way to express this is to assume one or more solutions or examples exists. Then there must be a smallest solution or example—a minimal counterexample. We then prove that if a smallest solution exists, it must imply the existence of a smaller solution (in some sense)—which again proves that the existence of any solution would lead to a contradiction. The method of infinite descent was developed by Fermat, who often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation ''r''2 + ''s''4 = ''t''4 and proving Fermat's theorem on sums of two squares, which states that any prime ''p'' such that ''p'' ≡ 1 (mod 4) can be expressed as a sum of two squares (see proof). 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). == Number theory== 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''/2''E'' 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'')/2''A''(''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proof by infinite descent」の詳細全文を読む スポンサード リンク
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