The Perfect Number Journey

Mathematicians and non mathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). The smallest such example is 6, since 6 = 1 + 2 + 3. Such numbers are called perfect numbers.

The search for perfect numbers began in ancient times. The first three perfect numbers: 6, 28 and 496 were known to the ancient mathematicians since the time of Pythagoras (circa 500 BC).

Verify for yourself that the numbers 28 and 496 are in fact perfect numbers, by completing the table below. You may use a calculator to work out the answers.

Table 1:  The first three perfect numbers.

 
 

How to find perfect numbers?

Euclid (circa 300 BC), the famous Greek mathematician, devised a simple method for computing perfect numbers.

Begin with the number 1 and keep adding the powers of 2 (i.e. doubling the numbers) until you get a sum which is a prime number. A perfect number is then obtained by multiplying this sum to the last power of 2.

In the exercise that follows, you are going to use this method to determine the next two perfect numbers. The first few rows in the table demonstrate the calculations being carried out to compute the first three perfect numbers. Apply this technique now, and let's see how fast you can find the fourth perfect number.

Before you proceed to find the fifth perfect number, you may want to pause for a moment and take a closer look at the first four perfect numbers that have been obtained this way. Record any interesting observations you have made about these numbers, and try to predict:

1. the number of digits in the 5th perfect number,
2. the last digit of the 5th perfect number.

Compute the fourth and the fifth perfect numbers by following Euclid's prescription. The first few rows of the table demonstrate the calculations used.

You may be keen to go on and calculate the 6th perfect number. Before you do that, I would like to draw your attention to Table 2 again.

While you were adding up the powers of two in the previous exercise, did you observe any pattern in the sequence of sums obtained? It may be more obvious if we express each number in terms of powers of two, as we shall see in the exercise that follows.

(a) Express the first two columns of numbers from the previous table in terms of powers of two.

Table 3:  The sequence of sums in powers of two.

(b) Using the shortcut calculation you have just learned, compute the following sum:

(c) Generalization. Write down the formula for the sum of the first n terms of the series:

(d) What is the largest power of two whose digits can be displayed fully on your calculator?

(e) A story and a related exercise: 

The sequence of numbers you have just examined

are called Mersenne numbers. This is the name given to numbers which are one less than a power of 2.