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

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

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

For n=2,

2^{n}-1=2^{2}-1=3 ( prime)

2^{n-1}=2^{2-1}=2^{1}=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,

2^{n}-1=2^{3}-1=7 (prime)

2^{n-1}=2^{3}-1=2^{2}=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:

*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.**

**External References**

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