Mystical Math – The Magic of Perfect Numbers

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Just to reiterate – in number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.

You can easily see that 6 is a perfect number – it has divisors 1, 2 and 3 (not including 6) and:

6 = 1 + 2 + 3

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.

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:

If, for some n>1,  2^{n}-1 is a prime number, then:

2^{n-1}\times (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,

2^{n-1}\times (2^{n}-1) = 2\times 3=6

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,

2^{n-1}\times (2^{n}-1) = 4\times 7=28

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.

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….)

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