Fermat's Infinite Descent

Pierre de Fermat's method of infinite descent is beautifully 
illustrated by the proofs of the following two propositions in 
Number Theory.  These are essentially equivalent to Fermat's
own proof, circa 1640, that the area of a Pythagorean triangle
cannot be a square, which he describe to Huygens simply by
saying "if the area of such a triangle were a square, then there
would also be a smaller one with the same property, and so on,
which is impossible".


Proposition:  There are no integer solutions of x^4 + y^4 = z^2.

PROOF:  Suppose there are integers x,y,z such that x^4 + y^4 = z^2.  
This can be written as a Pythagorean triple (x^2)^2 + (y^2)^2 = z^2, 
from which it follows that y^2 = 2pq,  x^2 = p^2 - q^2, and 
z = p^2 + q^2.  Since 2pq is a square, we know that either p or
q is even.  Thus, from the Pythagorean triple x^2 + q^2 = p^2 we 
have x = r^2 - s^2, q = 2rs, and p = r^2 + s^2.  Also, since 2pq
is a square we can set q = 2u^2 and p = v^2.  

Now, since 2u^2 = 2rs, we have r=g^2 and s=h^2.  These, along with 
p = v^2, can be substituted back into p = r^2 + s^2 to give 
v^2 = g^4 + h^4, where v is smaller than z, contradicting the 
fact that there must be a smallest solution.


Proposition:  There are no integer solutions of x^4 - y^4 = z^2.

PROOF:  Suppose there are integers x,y,z such that x^4 - y^4 = z^2.  
This can be written as a Pythagorean triple (y^2)^2 + z^2 = (x^2)^2.
If z is even this implies  y^2 = p^2 - q^2,  z = 2pq, and 
x^2 = p^2 + q^2, where x and y are both odd, from which we 
have p^4 - q^4 = (xy)^2.  Therefore, the existence of a solution
with even z implies the existence of a solution of the original 
equation with odd z, so we need only prove that a solution with 
odd z is impossible.

Assuming odd z, the Pythagorean triple implies y^2 = 2pq,  
z = p^2 - q^2, and x^2 = p^2 + q^2.  Since 2pq is a square, we 
can set q = 2u^2 and p = v^2.  Also, from the Pythagorean triple 
x^2 = p^2 + q^2 we have  p = r^2 - s^2, q = 2rs, and x = r^2 + s^2.

Now, since 2u^2 = 2rs, it follows that r = g^2 and s = h^2.  These, 
along with p = v^2, can be substituted back into p = r^2 - s^2 to 
give v^2 = g^4 - h^4, where v is smaller than z, contradicting the 
fact that there must be a smallest solution.


Fermat used this general approach for a variety of proofs.  In a
letter to Carcavi describing his methods he wrote

  "As ordinary methods, such as are found in books, are inadequate
   to proving such difficult propositions, I discovered at last
   a most singular method...which I called the infinite descent.
   At first I used it to prove only negative assertions, such as
   'There is no right angled triangle in numbers whose area is a
   square'... To apply it to affirmative questions is much harder,
   so when I had to prove 'Every prime of the form 4n+1 is a sum
   of two squares" I found myself in a sorry plight (en belle
   peine). But at last such questions proved amenable to my methods."
                          -Quoted from Andre Weil's _Number Theory_