When the number of divisors is greater than the number itself

 

The fact that a prime can be written as a nontrivial product of two complex integers is certainly very surprising to many, it may even be shocking.

Surprising or not, the fact is true and it would be right to go around. Perhaps not all primes “betray” the ancient classic character of not divisible building blocks. It can be seen quickly. that every prime is a “traitor”. What's more. We see that every prime number can be written in surprisingly many ways as a nontrivial product. (By the way, it would be absolutely right for someone to despise the mention of surprise, since it really isn’t a mathematical concept).

Let's see an example! The number of nontrivial divisors of the popular 13 is… 14!

 

-13+0i, -3-2i, -3+2i, -2-3i, -2+3i, -1+0i, 0-13i, 0-1i, 0+1i, 0+13i, 2-3i, 2+3i, 3-2i, 3+2i

 

But if 13 is capable of this, why be surprised that 12 has 38 nontrivial divisors among complex integers. Now, ready for anything, let’s look at the number of nontrivial divisors of all natural numbers between 1 and 100! These:

 

3, 10, 6, 18, 14, 22, 6, 26, 10, 46, 6, 38, 14, 22, 30, 34, 14, 34, 6, 78, 14, 22, 6, 54, 34, 46, 14, 38, 14, 94, 6, 42, 14, 46, 30, 58, 14, 22, 30, 110, 14, 46, 6, 38, 46, 22, 6, 70, 10, 106, 30, 78, 14, 46, 30, 54, 14, 46, 6, 158, 14, 22, 22, 50, 62, 46, 6, 78, 14, 94, 6, 82, 14, 46, 70, 38, 14, 94, 6, 142, 18, 46, 6, 78, 62, 22, 30, 54, 14, 142, 30, 38, 14, 22, 30, 86, 14, 34, 22, 178

 

I definitely have a feeling that we still have a lot of interesting things to do here.

 

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