Perfect Numbers (3/26) Definition. A number is called perfect if it is equal to the sum of its proper divisors. Examples: 6 and 28 are the first two (check!). Question. Are there infinitely many? Answer. Unknown. Question. Is there some “machine”, i.e., formula, which will crank out at least some perfect numbers? Answer. Yes, Euclid strikes again! Euclid’s Theorem on Perfect Numbers. If both p and q = 2 p – 1 are prime (i.e., q is a Mersenne prime), then 2 p – 1 (2 p – 1) = 2 p – 1 q is a perfect number.
Examples and Proof of the Theorem In the theorem, set p = 2. Then q = 3 and we get 2(3) = 6. Now set p = 3. Then q = 7 and we get 2 2 (7) = 28. So what does p = 5 generate? Remember, we must check that q = 2 p – 1 is in fact a Mersenne prime. Proof outline: Because q is prime, the proper divisors of 2 p – 1 q are: 1, 2, 4,..., 2 p – 1 ; q, 2q, 4q,..., 2 p – 2 q. By (again!) the summing of a geometric series, the sum of the first half above is 2 p – 1 = q, and the sum of the second half is q(2 p – 1 – 1). Now add those up! QED!
Other Perfect Numbers? Question. Is every perfect number of this form? Answer. Not known, but “sort of”, in the following sense: First, no one has ever found an odd perfect number, but also no one has ever proved that they can’t exist. That aside, however, Euler settles the matter: Euler’s Theorem on Perfect Numbers. Every even perfect number is of the form 2 p – 1 (2 p – 1) where p and 2 p – 1 are prime. The proof is, as you might expect, not simple.
The sigma Function We define a new number theory function, the sigma function (n). It is the sum of all the divisors of n (including 1 and n). Keep this straight from the Euler-Phi function. (What’s it measuring again?) Hence n is perfect if and only if (n) = 2n. Facts about this function: If p is prime, then (p k ) = 1 + p p k = (p k +1 – 1)/(p – 1) The sigma function is multiplicative (recall this term!). Example. What’s (300)?
Assignment for Friday Read Chapter 15 through page 105 (further if you like). Do Exercises15.2, 15.3 a & b, and 15.5