Math & Science Home  Proficiency Tests  Mathematical Thinking in Physics  Aeronauts 2000 

Perfect Numbers  A Case StudyPerfect numbers are those numbers that equal the sum of all their divisors including 1 and excluding the number itself. Most numbers do not fit this description. At the heart of every perfect number is a Mersenne prime. All of the other divisors are either powers of 2 or powers of 2 times the Mersenne prime. Let's examine the number 496  one of the known perfect numbers. In order to demonstrate that 496 is a perfect number, we must show that 496 = (the sum of all its divisors including 1 and excluding 496) We might just start by dividing and working out the divisors the long way. Or, we might begin by noting that, in the notation that includes a Mersenne prime, 496 = 24 (2^{5}  1) = 2^{4} x 31. From this expression, we may easily obtain the divisors of 496, namely: 1, 2, 2^{2}, 2^{3}, 2^{4}, 31, (2×31), (2^{2}×31), (2^{3}×31), and (2^{4}×31). The sum of all the divisors excluding 496 is then 1 + 2 + 2^{2} + 2^{3} + 2^{4} + 31× (1 + 2 + 2^{2}+ 2^{3}) In order to break this mess down a bit, let us examine the partial sum u = 1 + 2 + 2^{2} + 2^{3} + 2^{4} If we multiply by 2 (i.e., 2u = 2 + 2^{2} + 2^{3} + 2^{4} + 2^{5}) then subtract, we find u = 2u  u = 2^{5}  1 = 31 Thus, the sum of the divisors becomes = 31
+ 31(2^{4}  1) as we were to show.
n_{P} = 2^{c}(2^{c+1}  1) be a perfect number with (2c+1 1) being the embedded Mersenne prime. Then the divisors of n_{P} are 1, 2, 2^{2}, ... , 2^{c}, 2^{c+1}  1, 2(2^{c+1}  1), ..., 2^{c}(2^{c+1}  1) The sum of all the divisors excluding np is then 1 + 2 + 2^{2} + ... + 2^{c} + (2^{c+1}  1) (1 + 2 + ... + 2^{c1}) Again, we examine the partial sum: u = 1 + 2 + 2^{2} + ... + 2^{c}. We multiply by 2 (i.e., 2u = 2 + 2^{2} + ... + 2^{c} + 2^{c+1}) then subtract: u = 2u  u = 2^{c+1}  1 The sum of the divisors becomes: = (2^{c+1}
 1) + (2^{c+1}  1)(2^{c}  1) as we were to show.


Please send suggestions/corrections to: Web Related: David.Mazza@grc.nasa.gov Technology Related: Joseph.C.Kolecki@grc.nasa.gov Responsible NASA Official: Theresa.M.Scott (Acting) 