About the Cases in Which the Formula x^4+mxxyy+y^4 Can be Reduced to a Square
Year: 2024
Type: article
Abstract: Euler continues a previous study on the title Quartic (E696) with a new approach. His starting point here is the observation that, when the title Quartic is solved for m, the resulting fraction becomes an integer when z=ax^2y^2-(x^2±y^2). He provides many quadratic forms for m that allow special solutions, and tables for |m|≤200. Even though the tables are known today to be incomplete, they allow an insight into the enormous amount of work that was needed for their compilation. The interest in the title Quartic predates Euler, and the fact that he resumed the study indicates a certain level of dissatisfaction with his earlier results. He was interested in positively identifying values for m that allow solutions, and his methods enabled him to find many. However, difficult cases such as m=±85 escaped him because they required too large of a search range in parameter space. On the other hand, Euler was very much aware that his methods were not suitable for proving the nonexistence of solutions for given m, and that his results remained incomplete.
Source: Euleriana
Author: Georg Ehlers
Institution: Oak Ridge National Laboratory
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4402552593