Conjecture on n-sided Numbers and Their Sums Sep 10, 2021 | 1 minute

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


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.

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About me

Welcome! I’m Floyd Hightower.

I am a programmer who is passionate about making the world a better place using technology.


Life Principles

Here are some of the principles I try apply to every area of my life:

  • Let your ideas see the light of day
  • Always be willing to accept feedback and criticism (even when it is poorly delivered)
  • Rome ne s’est pas faite en un jour (Rome wasn’t built in a day)
  • Don’t be too proud to follow a good example
  • Loose ends always unravel
  • Unanswered questions never go away
  • The Tisroc won’t live forever whether you want him to or not
  • Talk less; listen more
  • Use more semi-colons
  • Use oxford commas
  • Learn how to politely say “No”
  • Don’t let the possibility of failure scare you away from starting something
  • To do something, you have to do something
  • A whiteboard is worth a thousand laptops
  • Ideas have consequences