The Euler-Fermat Theorem• Our text calls this “Euler’s Formula”, but I prefer the above

name, giving due credit to Fermat.• Obvious question: Can Fermat’s Little Theorem be generalized to

an arbitrary modulus m? The answer is yes!• Look back at the proof of the key lemma and of flt. As long

as GCD(a, m) = 1, the whole proof goes through until the very end. But when m is composite, (m – 1)! and m are NOT relatively prime, so we can’t cancel the (m – 1)! .

• How to fix: If we restrict ourselves to the numbers between 1 and m which are relatively prime to m, a very similar proof should now go through.

The Euler Phi-function• So, how many numbers are there between 1 and m which

are relatively prime to m? We will learn (pretty soon) how to compute this number fairly easily even for relatively large m, but for now we’ll just give this number a name:

• Definition. The Euler Phi-function, denoted (m), is the numbers of numbers between 1 and m which are relatively prime to m.

• So, for example (10) = 4, (9) = 6, etc., and of course,(p) = p – 1 if p is prime.

• Now rerun the proof of flt, using only the numbers which are relatively prime to m, of which there are (m).

• Now how many a’s are there in the left hand side?

The Theorem• The Euler-Fermat Theorem. If GCD(a, m) = 1, then

a (m) 1 (mod m).• Note that this is a direct generalization of flt.• Note that the condition that GCD(a, m) = 1 is necessary.

For example, is 24 1 (mod 10) true or false?• This theorem says then that in a congruence (mod m),

we can reduce any exponents by (m).• Examples. Solve x 2738 (mod 10).• Solve x43 25 (mod 9).• For Friday: 1. Work on Hand-in #3.

2. Read Chapter 10 and do Exercise 10.2.