#### Title

Euler's Early Work in Diophantine Equations

#### Document Type

Oral Presentation

#### Location

SURC 301

#### Start Date

17-5-2012

#### End Date

17-5-2012

#### Abstract

We examine the mathematical and historical context of Leonhard Euler’s first paper on Diophantine Equations, “De Solution Problematum Diophanteorum Per Numeros Integros,” (E29). We reexamine and verify Euler’s calculations, and we translate his work into modern notation. Euler struggled in working with Diophantine Equations at first, which makes his work difficult to follow. In fact, his difficulties are made more evident by the fact that we found several previously-unreported errors in the paper. We show how these errors can be fixed without changing the main idea of Euler’s argument. We also compare E29 to another paper on Diophantine Equations which Euler wrote late in his life. In this second paper, Euler’s mathematical ideas are much easier to understand and to verify, and his work is more complete, demonstrating that he had progressed in his understanding of this type of problem. In order to put the paper in context, other problems in Diophantine Equations such as solutions to the Pell Equation and Fermat’s Last Theorem are traced back to their roots and followed to their completion, and Euler’s life is examined briefly.

#### Recommended Citation

Livingston, Ben, "Euler's Early Work in Diophantine Equations" (2012). *Symposium Of University Research and Creative Expression (SOURCE)*. 147.

https://digitalcommons.cwu.edu/source/2012/oralpresentations/147

#### Additional Mentoring Department

Mathematics

Euler's Early Work in Diophantine Equations

SURC 301

We examine the mathematical and historical context of Leonhard Euler’s first paper on Diophantine Equations, “De Solution Problematum Diophanteorum Per Numeros Integros,” (E29). We reexamine and verify Euler’s calculations, and we translate his work into modern notation. Euler struggled in working with Diophantine Equations at first, which makes his work difficult to follow. In fact, his difficulties are made more evident by the fact that we found several previously-unreported errors in the paper. We show how these errors can be fixed without changing the main idea of Euler’s argument. We also compare E29 to another paper on Diophantine Equations which Euler wrote late in his life. In this second paper, Euler’s mathematical ideas are much easier to understand and to verify, and his work is more complete, demonstrating that he had progressed in his understanding of this type of problem. In order to put the paper in context, other problems in Diophantine Equations such as solutions to the Pell Equation and Fermat’s Last Theorem are traced back to their roots and followed to their completion, and Euler’s life is examined briefly.

## Faculty Mentor(s)

Dominic Klyve