# Solitary 10

Connections between integers are deeply studied in the field of Number Theory.  In certain instances, these abstractions are attributed character that takes shape in tangible analogs. When referring to colleague and prodigy, Srinivasa Ramanujan, the mathematician John E. Littlewood remarked, “Every positive integer was one of his personal friends.”

While these words speak of the uncanny ability Ramanujan had in seeing patterns and connections between numbers, the concept of friendship amongst numbers is a topic seriously considered by specialists in mathematics.  In particular, number theorists have created a rule that defines whether any two natural numbers are friendly.  Of course, friendship here is not the same as the classical understanding derived from the social interaction between humans.  Moreover, it is simply a rule carried out on two natural numbers to evaluate if certain conditions are met.  If any two natural numbers meet these conditions, they are deemed a friendly pair.

Rules in mathematics often involve the concept of a function.  Functions can be loosely thought of as a machine that receives an input and produces an output.   If an object is placed in the input, either the same object or a different object comes out of the output.  However, it is not considered a function if two different objects can come out of the output for one object in the input.

The divisor function is a function that takes a natural number n and sums the divisors of n, denoted $\sigma(n)$.   For example, {1,2,4} are the divisors of the natural number 4, because each of 1, 2, and 4 divide 4.  Therefore , $\sigma(4) = 1 + 2 + 4 = 7$.   Let’s further define the divisor function divided by n, $\sigma(n)/n$, known as the abundancy function denoted by $\sum (n)$.

With these definitions in hand, two natural numbers $m$ and $n$ are said to be a friendly pair, ($m$,$n$),  if $\sum(n)$ = $\sum (m)$.  Many pairs of numbers are known to be friendly pairs.   However, there are a few, with 10 being the least, in which there is no other known natural number $m$ such that 10 and $m$ are a friendly pair.   So, one can ask: Is 10 a solitary number?  Of course, solitary here means there does not exist a natural number $m$ in which $m$ and 10 are a friendly pair.

Showing that a number is not solitary can involve some rather large numbers.  For instance, 24 is not a solitary number, because 24 and the number 91,963,648 are a friendly pair.  Determining whether or not 10 is a solitary number is of great interest to number theorists and math hobbyists alike. Elegant approaches are yet to be found.  However, the solution may come from a novel use of known connections etched permanently in the history of mathematics.