Number Theory Section Summary: 11.2

Fermat's Last Theorem

1. Summary

   So we left things at all solutions of

    

   which can be written as

   

   for integers s>t>0 such that gcd(s,t)=1 with . In particular, there ARE integer solutions of that equation (1); so what about

   

   One observation is that, if n=pq, then

   

   and

   

   so that we simultaneously have solutions for all powers which are factors of n. Thus it suffices to ask if we can solve

   

   for primes p: if we can't solve it for the prime factors of n, then we can't solve it for n itself.

   Since we CAN find solutions for p=2, it's certainly possible that we have solutions for , for . Fermat, however, took care of that....

   Andrew Wiles recently (1994) proved that no solutions in integers exist for any power n greater than 2. In this section, we see how Fermat (who professed to have a proof of this theorem) solved the case of n=4.

2. Theorems

   Theorem 11.3: The Diophantine equation has no solution in the positive integers x, y, and z.

   Proof: by Fermat's method of ``infinite descent'': one obtains from a triple a strictly smaller triple, and so on ad infinitum; but the positive integers cannot be reduced ad infinitum - contradiction!

   Corollary: The equation has no solution in the positive integers x, y, and z.

   Corollary: The equation has no solution in the positive integers x, y, and z.

   Hence, the only exponents of interest left to prove are odd primes....

   Theorem 11.4: The Diophantine equation has no solution in the positive integers x, y, and z.

3. Properties/Tricks/Hints/Etc.
   • Fermat (1637) writes

     ``It is impossible to write a cube as a sum of two cubes, a fourth power as the sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have discovered a truly wonderful proof, but the margin is too small to contain it.''

     Fermat proved the case n=4, and hence n=4k.

   • Euler (1770) proved the result for the case p=3;
   • Dirichlet and Legendre (1825) independently proved the case p=5;
   • Lamé (1829) proved the case p=7;
   • Kummer (mid 1800s) proved the result for a large class of primes p (called the regular primes);
   • Faltings (1983) proved that all powers n>2 could have only finitely many triples as solutions; and
   • Andrew Wiles (1994) proved the whole enchilada....