2. An investigation of perfect numbers Our goal is to give a proof of the following: Theorem: Suppose n is an even perfect number, then n has the form n = where 2m – 1 is a prime. 2m-1(2m –1), (a) In class we proved that (rk * U)(n) = ok(n), showing that the arithmetic function Ok is multiplicative. Deduce a formula for o6(p°) where p is a prime and e is some natural number. Hint: List all the divisors of pº, notice that the sum of their kth powers forms a geometric summation and recall that a geometric summation has closed form gl+1 asi = a S - 1 j=0

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.