Magic Numbers
Problem Statement/Problem setup
We were instructed to search websites that give information about the history of
numeration. We chose five interesting pieces of information to bring back to the
group. One of the things I found concerned perfect numbers. In the information
along with Euclid, and Plato, there was a paragraph that read as follows: 4
perfect numbers from ancient times are: 6, 28, 496, 8128.
There is no record of these discoveries.
6=1+2+3
28= 1+2+4+7+14
496= 1+2+4+8+16+31+62+124+248
8128= 1+2+4+8+16+32+64+127+254+508+1016+2032+4064
Plans to Solve/Investigation/Exploration of the Problem
The class wanted to figure out why
these numbers are "perfect numbers." So, everyone began adding, multiplying,
etc. to try and find a pattern or patterns. After a few minutes of deliberation,
the group figured it out with the guidance of our instructor. The group finally
discovered that "perfect numbers" must be numbers where the proper factors (all
factors of a number except the number itself) add up to the number. Our
instructor put us on the right track when she told us that she figured out that
for the 6, the three numbers 1, 2 and 3 are the proper factors of 6. She then
figured this concept into all of the other perfect numbers. Otherwise we would
all still be trying to understand why these are called "perfect numbers!"
Extensions of the Problem
Extensions of this problem could be
looking into what abundant numbers, deficient numbers and other types of numbers
are. We discussed these as a class. I find this to be a fascinating concept. I
wonder if there are connections elsewhere, e.g., in nature, to these "perfect
numbers."
Author & Contact
Pam Joseph
prj1@aol.com
