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Perfect Numbers (3/26)

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.

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Perfect Numbers (3/26)

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  1. 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 andq = 2p – 1 are prime (i.e., q is a Mersenne prime), then 2p – 1 (2p – 1) = 2p – 1 q is a perfect number.

  2. 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 22(7) = 28. • So what does p = 5 generate? Remember, we must check that q =2p – 1 is in fact a Mersenne prime. • Proof outline: • Because q is prime, the proper divisors of 2p – 1 q are:1, 2, 4, ..., 2p– 1 ; q, 2q, 4q, ..., 2p – 2 q . • By (again!) the summing of a geometric series, the sum of the first half above is 2p – 1 = q, and the sum of the second half is q(2p – 1 – 1). Now add those up! • QED!

  3. 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 2p – 1 (2p – 1) where p and 2p – 1 are prime. • The proof is, as you might expect, not simple.

  4. 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 (pk) = 1 + p + ... + pk = (pk+1– 1)/(p – 1) • The sigma function is multiplicative (recall this term!). • Example. What’s (300)?

  5. Assignment for Friday • Read Chapter 15 through page 105 (further if you like). • Do Exercises15.2, 15.3 a & b, and 15.5

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