Number Theory Section Summary: 7.1-7.3

Euler's Phi Function

1. Summary

   Now we put Euler's phi function to work, generalizing Fermat's Little Theorem.

2. Theorems

   Theorem 7.5 (Euler): If and , then .

   Proof (first form, requires the following lemma):

   Lemma: Let n > 1 and . If , ,..., are the positive integers less than n and relatively prime to n, then , ,..., are congruent modulo n to , ,..., in some order.

   Corollary: Fermat's Little theorem!

   Proof (second form, which doesn't require the lemma, but which relies on Fermat's theorem):

   Lemma: If p|a, p prime, then

   

   for .

   Proof (by induction, using the Binomial theorem and Fermat):

   Proof of the theorem: