1. Proof that 10 is Solitary
Dan Heflin

2. What is a Solitary Number?
A solitary number is a number that has no friendly pair. That is, a solitary number is in a singleton group, with only itself.

3. What is a Friendly Pair?
A friendly pair is a pair of numbers that has the same ratio of the sum of divisors to the number itself.
So, for example, 6 and 28 are friendly.
6 has divisors 1,2,3,6; = 12/6 = 2
28 has divisors 1,2,4,7,14,28; = 56/28 = 2

4. Friendly Numbers
6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

5. Solitary Numbers
These numbers include all prime numbers, all powers of prime numbers, and all other numbers where the ratio of the sum of the divisors to the number itself matches no other ratio.
18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 are all solitary numbers.

6. Is 10 a Solitary Number?
Well, Mathematicians all around the world have wondered if 10 was a solitary number, or if it has a friend out there somewhere.
It is also the smallest number where people are unsure of whether or not it has a friend.
10 must either be a solitary number, or its friend has a very large index.

7. Our Proposition
The question at hand has been pondered for many years throughout Mathematical history.
The problem is, 10 is not a power of a prime, nor is it prime itself.
This means that in order to prove that it is indeed solitary, it must be disproved that it is friendly, or we must find a way to prove a number as being solitary.

8. However
No Mathematician has been able to prove or disprove this theory.
In order to do this, Mathematicians would have to discover a way to find solitary numbers, or discover a way to find friendly numbers.
But, no one has been able to do this.

9. The End
As a class we could sit down for a long time and try to find 10 a friend, but I think we would all agree not to do that!