How many perfect people do you know? Or, better, how many perfect people have lived in this world of human beings? Not many, I’m sure you all would agree. Well, the same is true, in a relative sense of the word, in the world of numbers. The ancient Greeks, who held a great reverence for the mysticism of numbers, had a certain category of numbers called “Perfect Numbers“. A perfect number is nothing more nor less than a positive integer whose proper divisors have a sum of the number itself. (It should be understood here that while any number is a divisor of itself — that is, 12 is a divisor of 12 — that’s considered to be an improper divisor. And so it is not used in this activity.) Now it just so happens, that like the scarcity of perfect people, there are not many perfect numbers, hence they become special and worthy of our attention.
The first, and therefore smallest, perfect number is 6. Its proper divisors are 1, 2, and 3. Their sum is, right, 6. The next perfect number is 28, as its proper divisors are 1, 2, 4, 7, and 14. Again, the sum of those integers is 28.
Simple as it now appears, why did I state that there are so few of this kind of number? Well, it’s because the 3rd one does not appear until nearly 500; the 4th one is over 8,000; and the 5th one is over 33 million!!! Does that put things into perspective now? You just don’t bump into a perfect number every day, at least those of the larger size.
It is, in part, for this reason that I’m always pleased to point out to my friends, collegues, and students a curious fact about my personal life. My wife’s name is Gloria, and mine is Terrel. Both names contain 6 letters. But wait, the best is yet to come. We were married on June 28. Note that June is the 6th month, so the digit form of that date is (ta-dah!) 6/28. Two perfect numbers.
In fact, the only two perfect numbers small enough to form a date in the calendar. In other words, 6/28 might be considered as the “perfect date” for a mathematician to get married!
- There are only 37 known perfect numbers! Most of them are so large that it would take many, many papes of paper just to write one such number. The number of digits for the 37th one, just recently found by the way (1998) is 1,819,050. Here is the way we can represent it via exponents: 23021376 × (23021377 – 1)[Note: See the formula in the heading of this article.]
- All known perfect numbers are even. No odd ones have been found as yet. But this is no proof that none exist.
You should be asking yourself the question: what about all those other numbers? How do they fit into the framework of the Greek mind? The answer is quite straightforward, actually, if you think about it. There are 3 things that can happen with the sums of the proper divisors of numbers: they can be equal to the number itself, as just shown above, or they can be lesser, or they can be greater, than the number itself. Some examples will illustrate what we mean by this:
- Take 10. Its proper divisors are 1, 2, and 5. Sum: 8.
- Take 12. Its proper divisors are 1, 2, 3, 4, and 6. Sum: 16.
There you have it. The Greeks called numbers that fall into category 1 as deficient, and those that fall into category 2 as abundant. The names are really quite self-descriptive, don’t you think?
Now armed with this new knowledge, you are ready to classify any number as deficient or abundant. Or if you are lucky, perfect.
In case you’re still wondering about the 3rd, 4th, and 5th perfect numbers alluded to above, I’ll give hints for the 3rd and 4th:
— one can be found between 490 and 500.
— the other can be found between 8,120 and 8,130.
The 5th one, however, is quite big, so I’ll just state it:
A reasonable math class challenge is to find all its divisors. With a
calculator it’s a snap!
For a listing of the 39 (it’s not 37 anymore!) Perfect Numbers, click HERE.