# There seem to be no one who can prove that the number ten isn’t a solitary number.

If you happen to be a math nerd, hold on to your seats, we’ve got a brain teaser for you. Can you prove that the number 10 is a solitary number? Spoiler alert: You just can’t. Proving that 10 is a solitary number has shown to be an impossible feat so far. However, what makes this so difficult?

### We’ve Gotta Be Friends

In attempts to prove that 10 is a solitary number, you need to really understand what we mean by a “solitary number” in the first place to begin with. Let’s take a crash course: A solitary number is a number that doesn’t have any friends. In other words, if a number has friends, it’s a “friendly number,” and if it doesn’t, it’s a solitary number. You’re now asking, what’s even a friendly number?

Now, a friendly number is a number that shares a relationship with another number — either in a friendly pair or friendly n-tuple. How cute. You and your friend might share a relationship over your mutual love for Marvel movies, but friendly numbers share a relationship over a mutual abundance index. To find a number’s abundance index, you add the sum of the number’s factors —  the numbers that are divisible by the number — and divide that sum by the number you started with.

### Lonely, I Am So Lonely, On My Own

Let’s take 6 as an example. Add the factors of 6, that is 1 + 2 + 3 + 6, the sum of 12. Now divide the sum by the original number 6. Voila, your abundance index is the number 2. The numbers 6 and 28 are a friendly pair because they both have a mutual abundance index of 2. (Here’s how 28 looks like: 1 + 2 + 4 + 7 + 14 + 28 = 56, 56 by 28 = 2).

(Fun fact: All perfect numbers are friendly and have an abundance of 2, and 6 is the smallest friendly number. Got it? Good!).

Wait a second, not all numbers are friendly. As a matter of fact, all prime numbers and prime powers are solitary numbers, which means they have no friends. In other words, running that little divisor exercise on solitary numbers will give you an abundance index that’s unique.

Now, let’s get back to the original problem: The number 10. It seems like the number upon which the entire metric system is based, would be an easy task. Weirdly, it hasn’t been that easy of a task to find another number that share it’s abundance index of 9/5. (Here’s how 10 looks like: 1 + 2 + 5 + 10 = 18, 18/10, or 9/5). At the very least, we don’t know for sure if 10 has any friends that are less than 2,000,000,000 (two billion). So, that’s a start? But 10 is definitely not alone in this unsolvable group.

### You’re Not Alone

Here’s a set of numbers believed to be solitary:

{14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others … }

Though proving it appears to be an extremely difficult task. Proving that 10, or any of the others, is a solitary number is what’s called an open problem, an unsolved problem in mathematics. And making matters even more complicated, there are only a handful of numbers.

Here’s a set of numbers known to be solitary:

{18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369, and many others … }

Why is this so hard? Well, proving that 10 has no friends is sort of trying to prove that unicorns don’t exist. So easy!  As far as you don’t come across one, you’re done, but not so easy to prove it right. Considering the infinite number of numbers out there — that’s not just impossible.