Method of infinite descent

[psi20]

How does the logic for the method of infinite descent work? Fermat indirectly proved that x^4 + y^4 = z^4 has no solutions through this.

"In order to prove that there were no solutions, Fermat assumed that there was a hypothetical solution (A,B,C). By examining the properties of (A,B,C), he could demonstrate that if this hypothetical solution did exist, then there would have to be a smaller solution (D,E,F). Then by examining this solution, there would be an even smaller solution (G,H,I), and so on. Fermat had discovered a descending staircase of solutions, which theoretically would continue forever, generating ever small numbers. However, x,y, and z must be whole numbers. So the never-ending staircase is impossible because there must be a smallest possible solution. This contradiction proves that the initial assumption must be false."

 

Something to that effect. But I don't get how it works. Can someone explain it to me?

[jcarlson]

Uh.

 

(0,1,1), (1, 0, 1), and (0, 0, 0) are all solutions to x^4 + y^4 = z^4... this is the equation of a surface in R^3, which looks like an inverted paraboloid...

[psi20]

x,y, and z can only be positive integers, forgot to mention that

[Lyssia]

"However, x,y, and z must be whole numbers. So the never-ending staircase is impossible because there must be a smallest possible solution. This contradiction proves that the initial assumption must be false."

 

I think it's not only that they must be integers but also that they must be positive integers; since the set of positives has a least element, infinite descent cannot happen and thus leads to a contradiction.

[matt grime]

It's a reductio ad absurdum idea.

 

Do you not understand how that works in general or in this particular example?

 

Here is a link explaining this particular example:

 

http://sweb.uky.edu/~jrbail01/fermat.htm