psi20 Posted May 11, 2005 Share Posted May 11, 2005 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? Link to comment Share on other sites More sharing options...

jcarlson Posted May 11, 2005 Share Posted May 11, 2005 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... Link to comment Share on other sites More sharing options...

psi20 Posted May 11, 2005 Author Share Posted May 11, 2005 x,y, and z can only be positive integers, forgot to mention that Link to comment Share on other sites More sharing options...

Lyssia Posted May 11, 2005 Share Posted May 11, 2005 "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. Link to comment Share on other sites More sharing options...

matt grime Posted May 11, 2005 Share Posted May 11, 2005 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 Link to comment Share on other sites More sharing options...

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