Abundant & Deficient
Numbers
So what happens when a number is not equal to the sum of its
divisors? Is it an imperfect number?
Now, we are going to explore what abundant and deficient numbers
are and how we can find them!
Abundant Numbers
So far, we have looked at perfect numbers. Now, we are
going to explore numbers that
are not perfect, but are still interesting! The first of which
are called abundant numbers.
In a mathematical context, a number is abundant if the
sum of its proper divisors (not including the number
itself) is greater than the number that you started with.
For example, the proper divisors of 12 are 1, 2, 3, 4 and
6. And 1 + 2 + 3 + 4 + 6 = 16 > 12. Therefore, 12 is
abundant.
In his book 'Introducto Arithmetica', Nicomachus states that he does not like abundant numbers.
He provides biological analogies in which he describes these numbers as deformed animals with
too many limbs, such as:“... ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or
having too many fingers on one of its hands....”
Deficient Numbers
Now let's look at the opposite of abundant numbers! We say that a
number is deficient if the sum of its proper divisors is less than the
number itself.
For example, the proper divisors of 10 are 1, 2 and 5. And 1 + 2 +
5 = 8 < 10.
Therefore, 10 is a deficient number.
Nicomachus did not have anything good to say about deficient numbers either. In his book,
deficient numbers are compared to animals with:“…a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have
a tongue...”
Activities
1. How many abundant numbers are there that are less than 100? What are they? What
do you notice about these numbers?
2. What is the first odd abundant number?
3. 45045 was the first odd abundant number to be discovered, by Carolus Bovillus.
Prove that 45045 is abundant.
4. Every number greater than 20,161 is the sum of 2 abundant numbers. Find an example
of this number and the 2 abundant numbers that add up to this specific number.
5. Identify a pair of abundant numbers with the same sum of proper divisors.
6. What do you notice about the multiples of abundant numbers?
7. How many deficient numbers are there that are less than 100? How did you figure this
out?
8. What do you notice about the divisors of deficient numbers?
9. What is the relationship between prime numbers and deficient numbers?
10. Now it is your turn! Create your own definition of proper divisors (e.g. you can say
that proper divisors do not include 1 and the number itself) and determine which
number less than 100 are abundant and deficient.