The Two Types of Numbers

There are two types of numbers. Solitary and Friendly/Amicable numbers.
The Solitary numbers don't have friends. Ok, it isn't exactly like that but it's similar.
To define a solitary number, we must first know what a friendly number is. A friendly number, also known as an amicable number, is a number in a friendly pair. The pair of numbers are so related that the sum of one of the number's proper divisors is the other number.
A Solitary number is not in a pair like that. There are many numbers that are thought to be solitary but proving it is hard.
It is hard to find a friendly pair. In fact, most numbers are solitary, including all primes and prime powers.
One of the famous unanswered questions is, is 10 a solitary number?
Well, if you can find a pair for ten, please comment and tell us.

next time: the math of rock paper scissors

Shinichi Mochizuki

Shinichi Mochizuki is a math genius. There's no doubt about that. Who is this Shinichi Mochizuki? Well, he claims to have proved the abc conjecture. Now, I say claims because while he turned in four papers for proof, no one can decipher them. 

Why could no one decipher these papers? Mochizuki has been working on the abc conjecture in solitude for so long that he has developed his own mathematical language which only he understands fully. 

Mochizuki posted his papers online one day. A couple of days later, a mathematician found it. The math community was buzzing excitedly. No one can decipher it yet. All we can do is wait, Mochizuki refuses to lecture about his papers to give us a better understanding. He has turned down several offers from great universities. All we can do is wait until a mathematician figures it out. Here are the papers:

Paper 1

Paper 2

Paper 3

Paper 4

I obviously cannot decipher them but if you feel like it, give it a go. 

Ok, the abc conjecture. What is it? 

Well, the abc conjecture starts off fairly simple. a + b = c. Now there are some restrictions. A and B cannot have any common prime factors. 

The abc conjecture also stated that the size of C is bound above by (roughly) the product of the distinct prime numbers dividing A, B, and C. 

For some better explanations:

The Business Insider: ABC Conjecture

As easy as 1, 2, 3, NOT!
If Mochizuki actually proved this conjecture, well, he has advanced his field more than 2 decades!
Figures. Mochizuki graduated high school at 16 and became a professor at age 33, which is unusually young. Here is his website:
The Thoughts of Shinichi Mochizuki

Let us see if any of you can make heads or tails of his writing. 

Next Time: Solitary Numbers and Friendly/Amicable Numbers