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 SEPTEMBER 2005

The Dean’s Math Column
Perfection in Mathematics

By Alfred S. Posamentier, Ph.D.

What is perfect in mathematics, a subject where most think everything is already perfect? Over the years various authors have been found to name perfect squares, perfect numbers, perfect rectangles, and perfect triangles. You might ask your students to try to add to the list of “perfection.” What other mathematical things may be worthy of the adjective “perfect?”

Begin with the perfect squares. They are well known: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . They are numbers whose square roots are natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .

A perfect number is one having the property that the sum of its factors (excluding the number itself) equals the number. The first four perfect numbers are:
6 [1+2+3],
28 [1+2+4+7+14],
496 [1+2+4+8+16+31+62+124+248], and
8128 [have your students find the sum of the factors].

They were already known to the ancient Greeks (Introduction Arithmeticae by Nichomachus, ca. 100 C.E.). Interestingly, the Greeks felt that there was exactly one perfect number for each digit-group of numbers. The first four perfect numbers seemed to fit this pattern, namely, among the single-digit numbers the only perfect number is 6, among the two-digit numbers there was only 28, then 496 was the only three-digit perfect number, and 8,128 was the only four-digit perfect number. Try asking your students to predict the number of digits in the next larger perfect number. No doubt they will say it must be a five-digit number. Furthermore, if you ask your students to make other conjectures about perfect numbers, they may conclude that perfect numbers must end in a 6 or an 8 alternately.

As a matter of fact, there is no five-digit perfect number at all. This should teach them to be cautious about making predictions with relatively little evidence. The next larger perfect number has 8 digits: 33,550,336. Then we must take a large leap to the next perfect number: 8,589,869,056. Here we also see that our conjecture (although reasonable) of getting alternate final digits of 6 and 8 is false.
*This is a good lesson about drawing inductive conclusions prematurely.

Perfect rectangles are those whose areas are numerically equal to their perimeters. There are only two perfect rectangles, namely, one having sides of length 3 and 6, and the other with sides of lengths 4 and 4.

There are also perfect triangles ** These are defined as triangles whose areas are numerically equal to their perimeters. Students should be able to identify the right triangles that fit that pattern by simply setting the area and perimeter formulas equal to each other. Among the right triangles, there are only the following two triangles, one with sides of lengths 6, 8, 10 and the other with sides of lengths 5, 12, 13.#
*The formula for a perfect number is:

If 2k—1 is a prime number (k > 1), then 2k—1(2k—1) is an even perfect number.

**See M.V. Bonsangue, Gannon, G. E., Buchman, E., Gross, N. “In Search of Perfect Triangles,” The Mathematics Teacher, Vol. 92, No.1, Jan. 1999, pp 56 – 61. #

Dr. Alfred S. Posamentier is Dean of the School of Education at City College of NY, author of over 35 books on math, and member of the NYS Standards Committee on Math. Read Math Wonders: To Inspire Teachers and Students. Alexandria, VA: ASCD, 2003. π: A Biography of the World’s Most Mysterious Number (Prometheus Books), 2004.