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.[2] It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.[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. 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, the original premise - that any solution exists - is incorrect: its correctness produces a contradiction.
An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example - a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then 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 earliest uses of the method of infinite descent appear in Euclid's Elements. A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.
The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation
r2+s4=t4
p\equiv1\pmod{4}
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. Since the terms are positive, they cannot decrease forever.
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.
The proof that the square root of 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. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. 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.[5] [6] [7] The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example .
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[8]). The following is an algebraic proof along similar lines:
Suppose that were rational. Then it could be written as
\sqrt{2}=
| p | |
| q |
for two natural numbers, and . Then squaring would give
2=
| p2 | |
| q2 |
,
2q2=p2,
so 2 must divide p2. Because 2 is a prime number, it must also divide p, by Euclid's lemma. So p = 2r, for some integer r.
But then,
2q2=(2r)2=4r2,
q2=2r2,
which shows that 2 must divide q as well. So q = 2s for some integer s.
This gives
| p | = | |
| q |
| 2r | = | |
| 2s |
| r | |
| s |
Therefore, if could be written as a rational number, then it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts, ad infinitum. But this is impossible in the set of natural numbers. Since is a real number, which can be either rational or irrational, the only option left is for to be irrational.[9]
(Alternatively, this proves that if 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.)
For positive integer k, suppose that is not an integer, but is rational and can be expressed as for natural numbers m and n, and let q be the largest integer less than (that is, q is the floor of). Then
\begin{align} \sqrtk&=
| mn\\ [6pt] | |||
|
\\ [6pt]&=
| m\sqrtk-mq | |
| n\sqrtk-nq |
\\ [6pt]&=
| \left(n\sqrtk\right)\sqrtk-mq | |||||
|
\\ [6pt]&=
| nk-mq | |
| m-nq |
\end{align}
The numerator and denominator were each multiplied by the expression ( − q)—which is positive but less than 1—and then simplified independently. So, the resulting products, say m′ and n′, are themselves integers, and are less than m and n respectively. Therefore, no matter what natural numbers m and n are used to express, there exist smaller natural numbers m′ < m and n′ < n that have the same ratio. But infinite descent on the natural numbers is impossible, so this disproves the original assumption that could be expressed as a ratio of natural numbers.
The non-solvability of
r2+s4=t4
q4+s4=t4
Suppose there exists such a Pythagorean triangle. 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. Primitive Pythagorean triangles' sides can be written as
x=2ab,
y=a2-b2,
z=a2+b2
\sqrt{yz}
b2
a2
b2
a2
a2
z=a2+b2
a
b
\sqrt{z}
a
b
\sqrt{z}
b
\sqrt{y}
a
a
z=a2+b2
In any of these cases, one Pythagorean triangle with two sides each of which is a square or twice a square has led to a smaller one, which in turn would lead to a smaller one, etc.; since such a sequence cannot go on infinitely, the original premise that such a triangle exists must be wrong.
This implies that the equations
r2+s4=t4,
r4+s2=t4,
r4+s4=t2
For other similar proofs by infinite descent for the n = 4 case of Fermat's Theorem, see the articles by Grant and Perella[11] and Barbara.[12]