Perfect Numbers

What are the Perfect Numbers?

Definition: A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128.

A Perfect Number “n”, is a positive integer which is equal to the sum of its factors, excluding “n” itself.

Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by,

N = 2^p-1(2^p -1) where p is a prime for which 2^p -1 is a Mersenne prime.

That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2^p -1 is a prime number. Undoubtedly Mersenne was familiar with Euclid’s book in coming up with his primes.

Perfect Number Table:

The following gives a table of the first nine Mersenne Primes and Perfect Numbers

History of Perfect Number

It is not known when Perfect Numbers were first discovered, or when they were studied, it is thought that they may even have been known to the Egyptians, and may have even been known before. Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers (O’Connor and Robertson, 2004).

The Pythagoreans found the number 6 interesting (more for its mystical and numerological properties than for any mathematical significance), as it is the sum of its proper factors, i.e. 6 = 1 + 2 + 3 This is the smallest Perfect Number, the next being 28 (Burton, 1980).

Though the Pythagoreans were interested in the occult properties of Perfect Numbers, they did little of mathematical significance with them. It was around 300 BC when Euclid wrote his Elements that the first real result was made. Although Euclid concentrated on Geometry, many number theory results can be found in his text (Burton, 1980).

We shall consider Euclid’s result in a moment, but first, let’s define Perfect Numbers more broadly. There are numerous ways to define Perfect Numbers, the early definitions being given in terms of aliquot parts. The author defines: A Perfect Number n, is a positive integer which is equal to the sum of its factors, excluding n itself.

Also Check: Euclidean Geometry

Solved Examples on Perfect Numbers

Example 1:

Find all the perfect numbers from 1 to 500.

Solution:

We know that every perfect number can be expressed as 2^{p – 1}(2^p – 1) where p is a prime number.

Using the above formula let us find the perfect numbers from 1 to 500.

For n = 2, 2^{2 – 1}(2² – 1) = 2(4 –1) = 2 × 3 = 6.

For n = 3, 2^{3 – 1}(2³ – 1) = 2²(8 – 1) = 4 × 7 = 28

For n = 5, 2^{5 – 1}(2⁵ – 1) = 2⁴(3² – 1) = 16 × 31 = 496

∴ the perfect numbers between 1 to 500 are 6, 28 and 496.

Example 2:

Check whether the following numbers are perfect numbers or not.

(i) 282

(ii) 8128

Solution:

(i) Factors of 282 are 1, 2, 3, 6, 47, 94, 141, 282.

The proper divisors of 282 are 1, 2, 3, 6, 47, 94, 141.

Now,

1 + 2 + 3 + 6 + 47 + 94 + 141 = 294 ≠ 282.

Thus, 282 is not a perfect number.

(ii) Factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128

Proper divisors of 282 = 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 

Now, 

1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

Thus, 8128 is a perfect number.

Example 3:

State true or false for the following statements:

(i) Every number can be expressed as 2^{p –1}(2^p – 1).

(ii) Till now there do not exist any odd perfect numbers.

(iii) The numbers which can be expressed as the sum of their proper divisors are called perfect numbers.

Solution:

(i) Every number can be expressed as 2^{p –1}(2^p – 1). – False.

(ii) Till now there do not exist any odd perfect numbers.  – True

(iii) The numbers which can be expressed as the sum of their proper divisors are called perfect numbers. – True.

Questions Based on Perfect Number

Question 1: Verify that 28 is a perfect number.

Question 2: Verify in the case 18 = 2 · 3 ² = p ^k q ^l that the sum σ(n) of all divisors satisfies the formula

σ(n) = (1+P+P²+…P^k)(1+q+q²+…q^l)

Video Lesson on Numbers