The Gauss Gazette: Early Mathematical Career

Early Mathematical Career

Carl Gauss was attracted to mathematics from an early age. In 1798 (at the age of 21), Gauss finished what would come to be recognized as a masterpiece of number theory, Disquisitiones Arithmeticae. This work set the foundation of modern number theory and consolidated earlier work done by Fermat, Euler, Lagrange, and Legendre and others. This earlier work was supplemented with original, insightful content provided by Gauss. This was the first book on number theory to bring the entire field together into one cohesive framework, as opposed to a collection of isolated observations and theorems. Disquisitiones also set a organizational convention that would later be mimicked by later books (and one that is still largely followed today). First, a theorem was stated formally, then the theorem was proved, then corollaries of the theorem were explored.

Despite the considerable amount of time and effort that Gauss invested in Disquisitiones, he still considered choosing a profession other than mathematics while he was at the University of Gottingen. In particular, he was deeply interested in philology (a field of study similar to linguistics). Another important consequence of Disquisitiones was that it introduced new helpful notation, such as the ≡ symbol for congruence. Finally, Gauss showed in Disquisitiones how a regular heptadecagon (a polygon with 17 equal sides) can be constructed with a compass and a straightedge. This problem had been unsolved since ancient Greece and Gauss's proof was the first progress in the field on constructible polygons (shapes that can be drawn using only a compass and a straightedge) in over 2000 years. Gauss was so proud of this accomplishment that he asked for a heptadecagon to be carved on his tombstone. This last wish was not granted because the engraver said it would be indistinguishable from a circle.

In 1799, Gauss finished his dissertation by discovering a new proof of the fundamental theorem of algebra (every polynomial of degree n, has n roots, counting multiplicity). Prior proofs of the theorem (done by Lagrange, Euler, Laplace, and others) were mostly correct but were technically not acceptable by today's standards. Their proofs assumed that every polynomial has n roots and showed that the form of these roots was a + bi. Gauss improved upon their work by not assuming the existence of roots. Gauss's proof was mostly geometric, and -- ironically enough -- is also technically incorrect due to topological reasons.

Another important discovery of Gauss's was that every number can be written as a sum of at most three triangular numbers. The proof of this result is beautiful and makes use of the result of a different proof. The proof of the triangular number theorem starts with the fact that every positive integer congruent to 3 mod 8 can be expressed as the sum of at most three squares.

$\color{white} \large Suppose $n > 0$ is a positive integer such that $n \equiv 3 \mod 8$, and suppose that $8n + 3$ can be written as the sum of three squares. \\ \\ By the congruence condition, each square must be odd. Let $a,b,c$ be positive integers. \\ \\ Then, \begin{align*} 8n + 3 &= (2a+1)^2 + (2b+1)^2 + (2c+1)^2\\ &= 4a^2+4a + 4b^2 +4b + 4c^2 + 4c + 3 \\ \implies 8n &= 4a^2+4a + 4b^2 +4b + 4c^2 + 4c \\ &= 4a(a+1) + 4b(b+1) + 4c(c+1) \\ \implies n &= \dfrac{a(a+1)}{2} + \dfrac{b(b+1)}{2} + \dfrac{c(c+1)}{2} \\ \\ \end{align*}$