The Community of Numbers}
M.V.N. Murthy, The Institute of Mathematical Sciences, Chennai

Whenever we think of the word community we always think of people. People coming together on the basis of some thing in common form a community. They may be just friends, they may be relatives, they may just be people happy to be together or simply solitary people who have no one to care. Some times it is used for colonies of ants and such other groups or families of living organisms. We normally use these adjectives happy, perfect, friendly, solitary to describe persons or groups of persons.
Did you know that numbers are some times referred to by these same colourful adjectives? While this may show that mathematicians, especially number theorists, have a sense of humour, the adjectives do have deep meanings.
So let us start from the beginning---the natural numbers, 0,1,2,3,... A study of these natural numbers and beyond is often called number theory. We all know that these may be divided into even numbers, numbers divisible by 2 and odd numbers which do not have 2 as a factor. Ofcourse all numbers have 1 as their divisor. 

Perfect Numbers
Among these natural numbers some are called rerfect numbers. What is perfect about them? Number theorists tell you that they are perfect since all the factors (except the number itself) add to give the number itself. For example 6 is a perfect number since its factors 1+2+3 add to give you 6 itself. Such numbers have been known for a long time, more than about 2000 years. Surprisingly, such numbers are very rare. In fact it was known early that 6, 28, 496, 8128 are the first four perfect numbers. It took a long time, almost 1500 years, to discover the next perfect number and it turns out to be a large number 33,550,336. In 1772 the famous mathematician Euler discovered a very large number, for his time, 2,305,843,008,139,952,128 as a perfect number. The largest such perfect number known today has more than 34 million digits and the list has less than 50 such perfect numbers. Rare indeed.
Notice that all these perfect numbers are even. Is there any perfect number that is odd? Are there infinitely many perfect numbers or is their number finite? These are some open questions for which we do not have answers yet.

Friendly Numbers
Then there are these Friendly Numbers. Take any number N. Collect all its divisors and construct a function S(N) which is the sum of all the divisors including the number itself, namely N. Two numbers N and M are friendly if the ratio S(N)/N=S(M)/M.
There may be more than two numbers that share this friendliness! This is called an equivalence relation and numbers having the same ratio belong to the same equivalence class or simply clubs. 
For example
S(6)/6=(1+2+3+6)/6=2
S(28)/28=(1+2+4+7+14+28)/28=2.
So we say 6 and 28 are friendly numbers. In fact all perfect numbers are also friendly numbers since they all yield S(N)/N=2 by definition of perfect numbers.
As an exercise you can check that 30 and 140 are friendly numbers since S(30)/30 =S(140)/140=12/5. You may also add 2480,6200 and 40640 to this club of friendly numbers since they also have share the same ratio 12/5. 

Solitary Numbers
Are there number that are not friendly. Indeed yes---these are simply called solitary numbers. Easy to guess. All prime numbers, which have no divisors except 1 and the prime number itself, are solitary. For example if p is prime number then S(p)/p=(1+p)/p, which is different for each prime number p. In fact, any number which is a power of prime is also solitary by the same token.
Are there other numbers which are solitary but not prime? It is conjectured that 10 is a solitary number but has not been proven. It is possible that it is not solitary but then its smallest friend may be a very large number but it is not known. Unfortunately, there is no known general method that can be used to determine a number, for example 10, as solitary or friendly except by actually searching for and finding it!

Amicable Numbers
Some times two numbers are said to be Amicable Numbers. These are different from friendly numbers though they also involve divisors of a pair of numbers. For example consider the pair of numbers 220 and 284: The sum of proper divisors of 220 is 1+2+4+5+10+11+20+22+44+55+110=284. Similarly the sum of proper divisors of 284 is 1+2+4+71+142=220.
We say that the pair (220,284) form an amicable pair. In fact this is smallest such pair. The next few are (1184,1210), (2620,2924),(5020,5564), etc. 
Such pairs of numbers were known from the time of Pythagoras and were thought to have some mystical properties. We now know that there are more than 12 million such pairs. In all the cases the numbers in the pair are both even or both odd. It is not known whether even-odd pair of amicable numbers exist.

Happy Numbers
If numbers can be perfect, friendly or solitary, can they also be happy
or unhappy? Indeed they can be provided they satisfy the following
happiness criterion or not. A number is called a happy number if sums of
squares of its digits added sequentially reduce ultimately a single
digit given by 1. Trivially 1 is a happy number since its square is
always 1. Let us take the next one 7: (SUPER) 72=49, 42+92=97, 92+72=130, 12+32+02=10, 12+02=1. So 7 is a happy number. You can easily check that 1,7,10,13,19, etc are all happy numbers. Happiness of a number is unaltered if you interchange the digits or add any number of zeros in between. 
You may think of variations on the happiness theme by using cubes instead of squares. For example you can check 1579 is a happy number with cubes replacing the squares. If you are adventurous you may even consider higher powers. However, you will soon realise that there are not many when you go for higher powers.

And many more
These are some examples of numbers which have some interesting properties. We give funny names to some of their properties. In fact there are more: For example you may have heard of Twin Primes which are prime numbers that differ by 2 like (3,5), (5,7), (11,13) etc. But there are also Cousin Primes which differ by 4 like (3,7), (7,11), (13,17) etc. If they differ by 6, they are called Sexy Primes.
Apart from such interesting names, some times there are some deep results hidden in these classifications. Number theory essentially deals with such problems related primarily to the study of natural numbers. This is one of the oldest branches of mathematics, which used to be called simply Arithmetic, which originated in many older traditions like the Babylonian, Chinese, Greek, and Indian.
In number theory, if a number belongs to a certain category or variety,then one immediate question that mathematicians try to ask is if there are finite or infinite numbers in that category. Some times there are conjectures whether a number belongs to one category or the other. For example, you may ask are there infinitely many clubs of friendly number just as you can ask are there infinitely many perfect numbers? This are still open problems though there may be many unproven conjectures.
In contrast to many other branches of Mathematics, the problems and theorems in number theory may be stated in such a way that it is understood by even lay persons. However, the solutions and proofs may require very sophisticated background and techniques. Often the solutions may indeed take decades or centuries some times.