# An Infinite Descent

What Is Infinite Descent?

Name required. Post to Cancel. The following is an algebraic proof along similar lines:.

• Updated at finite intervals.
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Then it could be written as. Then squaring would give.

Because 2 is a prime number , it must also divide p , by Euclid's lemma. But this is impossible in the set of natural numbers. So two resulting products, say m' and n' , are themselves integers, which are less than m and n respectively.

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The following more recent proof demonstrates both of these impossibilities by proving still more broadly that a Pythagorean triangle cannot have any two of its sides each either a square or twice a square, since there is no smallest such triangle: [7]. Suppose there exists such a Pythagorean triangle. Then it can be scaled down to give a primitive i.

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In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle. One typical application is to show. In mathematics, the method of infinite descent is a proof technique that uses the fact that there are a finite number of positive integers less than any given positive integer. In other words, there is no infinite sequence of strictly decreasing non-negative integers.

The property that y and z are each odd means that neither y nor z can be twice a square. Furthermore, if x is a square or twice a square, then each of a and b is a square or twice a square.

## Finite Arithmetic with Infinite Descent

There are three cases, depending on which two sides are postulated to each be a square or twice a square:. 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.