The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. -- G. H. Hardy

Here's to pure mathematics! May it never have any use -- G. H. Hardy

It often turns out that much fun can be had by defining new classes of  numbers or other mathematical objects and investigating their resulting

mathematical properties and relations.   -- L.R. Duffy

There are many classes of numbers that mathematicians study such as even, odd, and prime natural numbers.  There are also more special classes of numbers like perfect numbers.  A number n is said to be perfect if the sum of its proper factors add up to n.  For instance the proper factors of 6 are 1,2, and 3  and 6=1+2+3, hence 6 is perfect.  Or, an n-digit number is said to be narcissistic if the sum of the nth powers of its digits is n.   153 is narcissistic because 1³ + 5^{3 +} 3³ = 153.

In this project we will be looking at a similar class of numbers that , as far as I know, have no special name.  We define them in the following way:

Let S(n) represent the sum of the digits of n. For example:

S(2)=2

S(365) = 3+6+5 = 14

and so on.  Find all positive integers with the property that S(n³) = n.

1. Show there a finitely many of these numbers.
2. Find an upper bound
3. Generalize this problem in some interesting way.
4. Do you proofs of 1 and 2 generalize to your new set of numbers?
5. Make up your own class of numbers.
6. Prove something about them.