Fermat invented this method of infinite descent. The idea is: If a given natural number with certain properties implies that there exists a smaller one with these properties, then there are infinitely many of these, which is impossible.
Here is an example:
Suppose is a square. Then, since are coprime and of opposite parity, . Thus, for to be a square, each of must be squares itself. Setting , , we have .
We prove that the Diophantine equation has no solution in natural numbers.
Suppose that , where , . Then is odd, and have opposite parity.
If was even, then , so or . But conflicts with . And implies contradicting . Thus, is odd, and implies that have opposite parity. ∎
Suppose is odd and is even. Then we have , and , where have opposite parity and are coprime. Since is odd, this implies , so it is sufficient to show that there is no solution for odd .
Now are assumed odd. Then is even, and there exist , , such that
Since is a primitive Pythagorean triple, there exist , , satisfiying
Since is a square and are coprime and, say, is odd, is a square, and we have , .
From the primitive Pythagorean triple we get , , . Since is a square, and each of and is a square: , .
Substituting in we have . But since this implies , thus we have another solution with odd . This contradicts to the fact that there exists a smallest solution.
See http://mathpages.com/home/kmath144.htmhere for a discussion of infinite descent vs. induction.
|Date of creation||2013-03-22 14:07:56|
|Last modified on||2013-03-22 14:07:56|
|Owner||Thomas Heye (1234)|
|Last modified by||Thomas Heye (1234)|
|Author||Thomas Heye (1234)|