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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:
Let be coprime positive integers with opposite parity, , and, say, is even.
Let ,
,
. Then is a primitive Pythagorean triple, and the area of the right triangle with sides is
.
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.
Remark 1 Suppose that
, where
,
. Then is odd, and have opposite parity.
Proof. 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 here for a discussion of infinite descent vs. induction.
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