Math Concepts

Mystical Math – The Magic of Perfect Numbers

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Jan 21, 2021

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Introduction 

Prime numbers and their properties were first studied extensively by ancient Greek mathematicians.

The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.

A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.

A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.


What is a perfect number?

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The discovery of such numbers is lost in prehistory. Pythagoreans (founded c. 525 BCE) studied perfect numbers for their “mystical” properties.

 
important notes to remember
 
“Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect."

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Mystical Math – The Magic of Perfect Numbers

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What is the smallest perfect number?

The smallest perfect number is 6 – it has divisors 1, 2, and 3 (excluding 6) and:

6 = 1 + 2 + 3

Theory of Perfect Numbers

However, you need not worry beyond a point because about 2000 years ago, Euclid came up with a handy algorithm for finding the perfect numbers. Mathematically, his proposition can be stated as: Now, if you are worried about finding the next perfect number using only the definition, then you are not alone! It is indeed a daunting task.

If, for some n>1,  \(\begin{align}{2^{\{ n \} }-1}\end{align}\) is a prime number, then:

(2-1)*(2n-1) is a perfect number.

Now let’s start testing this algorithm starting with the value of n as 2.

For n=2,

2n-1=22-1=3 ( prime)

2n-1=22-1=21=2

Therefore,

\(\begin{align}\left( {{2^{n - 1}}} \right) \times \left( {{2^n} - 1} \right) = 2 \times 3 = 6\end{align}\)

We have already shown that 6 is a perfect number. So the algorithm works for n=2.

For n=3, 

2n-1=23-1=7 (prime)

2n-1=23-1=22=4

Therefore,

\(\begin{align}\left( {{2^{n - 1}}} \right) \times \left( {{2^n} - 1} \right) = 4 \times 7{\rm{ }} = 28\end{align}\)

Hence, according to the algorithm, 28 should be the next perfect number.

But we need to verify this using the definition of perfect numbers:

The divisors of 28 are 1,2,4,7 and 14.(Excluding the number itself)

And 28 can be written as:

28 = 1 + 2 + 4 + 7 + 14

Voila! 28 is indeed a perfect number.


So the algorithm works for n=3.

So here we are! We now have our 2nd perfect number which is 28. We used the algorithm to arrive at this number and verified that it is perfect by using the definition of perfect numbers.

Now, can you find the next two perfect numbers? (Pro Tip: Try those values of n which are prime; for example, when n is equal to 5, 7 11, or 13….)


Fun Facts about prime numbers

  • Prime numbers are often used in cryptography or security for technology and the internet.
  • The number 1 used to be considered a prime number, but it generally isn't anymore.
  • The largest prime number known has around 13 million digits! The Greek mathematician Euclid studied prime numbers in 300BC.

Conclusion

This blog mainly talks about what is perfect numbers and the theory of perfect numbers and few problems. Mathematically, even perfect numbers give a good number theory example to the general idea of classification, i.e. all even perfects have a specific form.


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Frequently Asked Questions (FAQs)

Is Zero a perfect number?

Zero is neither a natural number nor a positive integer, hence is not a perfect number.

Why is 10 the perfect number?

No, 10 is not a perfect number. The divisors of 10 are 1, 2, 5, and 10. Therefore, the proper divisors of 10, or the divisors of 10 other than 10   

What is the most interesting number?

Therefore the number 6174 is the only number unchanged by Kaprekar's operation — our mysterious number is unique. The number 495 is the unique kernel for the operation on three-digit numbers, and all three-digit numbers reach 495 using the operation.


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