Table of Contents
1. | Introduction |
2. | History of Perfect Numbers |
3. | Properties of Perfect Numbers |
4. | Even Perfect Numbers |
5. | Odd Perfect Numbers |
6. | Summary |
7. | Frequently Asked Questions (FAQs) |
September 6, 2020,
Reading Time: 3 mins
Introduction
A natural number equal to the sum of all its proper divisors is known as a perfect number. 6 is the first perfect number, it is the sum of 1,2, and 3. It is followed by 28 and 496. As you progress to larger numbers, perfect numbers become harder to find.
History of Perfect Numbers
Perfect Numbers were initially studied by Pythagoras in 525 BCE. Pythagoras studied them for their ‘mystical properties’. His work was built upon by Nicomachus of Gerasa around 100 CE. He further classified it into deficient, perfect, and superabundant.
Early Greek mathematicians only knew the first 4 perfect numbers. Pythagoras discovered a close link between ‘twoness’ and perfect numbers, i.e there is a relation between powers of the number 2 and perfect numbers. 2 centuries later Euclid would have to refine Pythagoras’ link.
A statue of Pythagoras
Downloadable PDF
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Perfect Numbers |
Properties of Perfect Numbers
Perfect numbers are numbers where the divisors of the number excluding the number itself, add up to give the number itself.
6, 28, 496, 8128, 33550336 are the first 5.
Euclid discovered that all perfect numbers are always a multiple of two numbers, one which is the power of 2.
2^{p -1} × (2^{p} -1)
2^{1 }× (2^{2 }− 1) = 2 × 3 = 6
2^{2} × (2^{3 }− 1) = 4 × 7 = 28
2^{4 }× (2^{5 }− 1) = 16 × 31 = 496
2^{6 }× (2^{7 }− 1) = 64 × 127 = 8,128
This is applicable only where p is a prime number.
Today’s computers have been able to compute a perfect number with over 130,000 digits.
Abundant Numbers
When the sum of the divisors of a number, excluding the number itself is greater than the number, the number is called an abundant number.
An example is 12.
Divisors: 1 + 2 + 3 + 4 + 6 = 16
Deficient Numbers
When the sum of all the divisors of a number, excluding the number itself is less than the number, the number is called a deficient number.
An example is 10.
Divisors: 1 + 2 + 5 = 8
Both terms are derived from Greek Numerology.
Amicable Numbers
Two numbers that have proper divisors, such that they add up to the other number are called amicable numbers.
An example is 284 and 220.
Semiperfect number
A semiperfect number is a number that is the sum of all or some of its proper divisors. Most abundant numbers are semiperfect.
An example is 20.
Divisors: 1 + 4 + 5 + 10 = 20
Even Perfect Numbers
Perfect numbers in the form 2^{p - 1}×(2^{p}− 1) are all even perfect numbers.
Mersenne Prime Numbers
Prime numbers in the form of 2^{p }− 1 are called Mersenne Prime Numbers, they were discovered by Marin Mersenne a monk of the seventeenth century.
2^{p }− 1 is only prime if p itself is a prime number, however, this is not true for all p where p is prime.
In the eighteenth century, Leonhard Euler proved that all even perfect numbers can be expressed as 2^{p - 1} × (2^{p }− 1) although it was first conjectured in 1000 AD.
There is a correlation between Mersenne primes and Even perfect numbers. Each Mersenne prime yields one perfect number and vice versa. This is called the Euclid-Euler Theorem.
Odd Perfect Numbers
There are no known odd perfect numbers. Jacques Lefèvre said all perfect numbers are expressed in the form of Euclid’s rule. Implying there are no odd perfect numbers.
Summary
This blog covered perfect numbers and their discovery, history, and properties. A lot of research has been done on perfect numbers, however, there still is no known odd perfect number. With powerful computers today we have discovered really large perfect numbers.
Frequently Asked Questions (FAQs)
What are Perfect Numbers?
Numbers which are equal to the sum of all their proper divisors are called Perfect Numbers
An example is 28.
Its divisors are 1 + 2 + 4 + 7 + 14 = 28
List the first 5 perfect numbers?
Here is a list of the first 5 perfect numbers:
6
28
496
8128
33550336
Which is the smallest perfect number?
The smallest perfect number is 6. 6 is a sum of 1,2 and 3.
How many perfect numbers are there?
There are 51 known perfect numbers, the largest one discovered in 2018 has 49,724,095 digits.
Are there any odd perfect numbers?
There are no known odd perfect numbers.