Perfect Numbers: Overview, Questions, Preparation

The world of mathematics is full of concepts with no direct use in nature or science, but are fascinating nevertheless. Numbers can create all sorts of exciting patterns and develop new concepts, definitions, and techniques. One such concept is the topic of perfect numbers.

A perfect number in mathematics is a number that is numerically equal to the additive of its divisors apart from itself. These divisors include both prime factors and non-prime factors, all taken just once. For example, the number 6 has factors 1, 2, 3, and 6. Now, the sum of all its factors except itself is 1 + 3 + 2 = 6. Hence, 6 may be called a perfect number. The sum of all the divisors of a number except itself is known as the number’s aliquot sum. Hence, a perfect number can be said to be a number that is equal to its aliquot sum.

Another example of a perfect number is 28. The number 28 may be expressed in the following ways: 1 x 28, 2 x 14, 4 x 7, 7 x 4, 14 x 2, and 28 x 1. Hence, the divisors of 28, except itself, are 1, 2, 4, 7, and 14. The aliquot sum of the number 28 is 28, making it a perfect number.

Euclid’s Elements is the world’s earliest complete treatise on mathematics. The concept of perfect numbers is also mentioned in Euclid’s Elements, referred to as a complete number. Euclid was also able to prove in his treatise a rule for even perfect numbers. For this rule, consider q to be a prime number of the form 2p - 1. This is called a Mersenne prime. Now, the number q(q+1)/2 is a perfect number. Euler was later able to prove that all even numbers have this in common. This theorem is hence called the Euclid Euler Theorem.

About the Topic

In the NCERT syllabus, perfect numbers is a rather short topic generally reserved only for short answers or multiple-choice questions. They are not used for medium or long answer questions. Generally, this topic will not appear for more than 1 or 2 marks in an exam.

1. Prove that 28 is a perfect number.
The divisors of 28, except itself, are 1, 2, 4, 7, and 14. The aliquot number 28 is 28, making it a perfect number.
2. Is 20 a perfect number?

The divisors of 20, except itself, are 1, 2, 4, 5, and 10. The aliquot sum of the number 20 is 22, making it an imperfect number.
3. Is 6 a perfect number?

For p = 2, 2p - 1 = 3. 3(3+1)/2 = 6. Hence 6 is a perfect number.

Q: What are the first few perfect numbers?

Q: How many odd perfect numbers are known?

Q: Which perfect number is the sum of two cubes?

Q: What is an aliquot sum?

Q: How can a perfect number be defined in terms of aliquot sum?