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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