Some properties of perfect numbers

The perfect numbers possess a number of interesting properties, some of which are easily observed but some need to be studied in greater depth.

One rather interesting property is that the sum of the reciprocals of all the divisors (including the number itself) of a perfect number is 2. For example, take the case of the perfect number 28:

This little observation may seem a bit surprising at first, but a closer study reveals that it follows directly from the definition of perfect number. 

Another observation is that perfect numbers seem to end in either 6 or 8, though not necessarily alternating between the two. The following exercise will examine why this is so.

Exercise 7

The table below should be of great help in the investigations that follow.


Table 5:  Patterns of terminating digits.

(a) What is the last digit of 21997?

(b) What are the last two digits of 21997?

(c) Explain why the last digit of a perfect number ends in either 6 or 8.

(d) Explain why the last digit of a perfect number ends in either 6 or 28.


 
 

How are perfect numbers & triangular numbers related?

We will now examine yet another property of perfect numbers. Look at the sequence of triangular numbers above. The perfect numbers 6 and 28 appear in the sequence, so they are also triangular numbers. What about the other perfect numbers, are they also triangular? We will find this out in the following exercise.
Exercise 8

(a) Find the 200th triangular number.

(b) A related story.

(c) Denote the nth triangular number by n. Write down the formula, in terms of n, for n.

(d) The perfect numbers 6, 28, 496 and 8128 are the 3rd, 7th, 31st and kth triangular numbers respectively. Find the value of k.

(e) Given thatt is the 8th perfect number, find the value of t.

*(f) Prove that all even perfect numbers are triangular.



 
 

Are all perfect numbers hexagonal?

Have you encountered hexagonal numbers? Like the triangular numbers, these numbers can be visually represented, as in the figure below.

I'll leave it to you to explore on your own the relationship between hexagonal numbers and perfect numbers. The following exercise may help.
Exercise 9

(a) Find, in terms of N, the Nth hexagonal number.

*(b) Prove that all even perfect numbers are hexagonal.