The Sum of Four Squares and Quaternions

Elizabeth Zeigler

Winner of the Judges’ Choice Award for Best Math Presentation

When looking at the set of natural numbers, we see that not every natural number is a perfect square or the sum of two squares with 3 being the first number this occurs. However, we can write 3 as the sum of three squares as shown: 3 = 1 + 2 = 1 + 1 + 1
Next, we look at 7, which cannot be written as the sum of three squares, but we can write 7 as the sum of four squares as shown: 7 = 1 + 6 = 4 + 3 = 1 + 1 + 5 = 1 + 4 + 2 = 1 + 1 + 1 + 4
To show all natural numbers can be written as the sum of four squares, we will use number theory techniques along with quaternions.

• Elizabeth Zeigler is a senior mathematic dual engineering major from Bonneau, SC. She has been a manager for the Lander University Softball Team and worked for Lander Athletics as a photographer/videographer. Elizabeth plans to attend Clemson in the fall of 2022 to pursue a civil engineering degree and work in the bridge construction industry.