While reading The Most Beautiful Mathematical Formulas: An Entertaining Look at the Most Insightful, Useful, and Quirky Theorems of All Time, I came across Lagrange’s four-square theorem:

We can represent every natural number as the sum of, at most, four squares.

For example, we can represent 27 as:

16 + 9 + 1 + 1

or

4^2 + 3^3 + 1^2 + 1^2

As I was reading about this, I came up with the following conjecture. This conjecture may already exist (I haven’t looked into the subject yet) or it may be false. From a cursory analysis, it seems to hold true.

My conjecture is:

We can represent every natural number as the sum of, at most, n n-sided numbers.

I need to formalize what I mean by “n-sided number”, but intuitively, I mean that, when n=3, n-sided numbers are triangular numbers; when n=4, we are dealing with square numbers; n=5 means pentagonal numbers; etc…

I plan to do some research in the future to see if this already exists and, if not, how I might go about proving it.