Proof of Euler-Fermat (2/28) Here’s an outline of the proof of the Euler-Fermat Theorem, which mirrors the proof of flt. Given any m, let B = {b i | 0 < b i < m and GCD(b i, m) = 1}. Note that B has  (m) elements. For any a  B, if we multiply all elements of B by a and then reduce by m, “The Scrambling Lemma” says we get B back, in a different order. Now multiply all together. Hence ab 1  ab 2 ...  ab  (m)  b 1  b 2 ...  b  (m) (mod m), i.e., a  (m) (b 1  b 2 ...  b  (m) )  (b 1  b 2 ...  b  (m) ) (mod m). Finally, we can cancel (b 1  b 2 ...  b  (m) ) from both sides because that product is relatively prime to m. The E-F Theorem follows.

Computing the Euler Phi-Function Because of the Euler-Fermat Theorem, it is clear that to solve congruences which have exponents larger than the modulus, we need to be able to compute  (m). Going through and checking numbers one by one for relative primeness to m is not a feasible algorithm. So let’s work out a formula for  (m). We use a very typical strategy in number theory: Settle what you want for prime numbers and their powers, then Use the Fundamental Theorem of Arithmetic to “patch together” the information about separate primes to get the result for any number.

Multiplicative Functions In number theory, the following definition is standard: Definition. A function f whose domain is integers (or a subset thereof), is called multiplicative if f(ab) = f(a)f(b) provided that a and b are relatively prime. If we know that a function is multiplicative and if we can figure out what it does to prime powers, then by the FTA we know what it does to arbitrary numbers. You shall read, and we shall discuss Monday, that the Euler-Phi function is indeed multiplicative. Hence it will suffice to figure out what  (p k ) is for any prime p and any positive k.

What’s  (p k ) ? Well, first, we know already that  (p) = p – 1 (= p 1 – p 0 ). As always, try out some examples: What is  (25) =  (5 2 )? Just list it out! What is  (27) =  (3 3 )? See any patterns? Conjecture: If p is prime,  (p k ) = ? Now prove it. Okay, assuming then that  is multiplicative, we can now easily compute  of any number which we know how to factor into its FTA form. Examples: What is  (100)? What is  (363)? What is  (720)?

For Monday Hand-in #3 is due at class time. Read Chapter 11. Do Exercises 11.1 and Have a good weekend, but......