From the algebraic methods of Diophantus–Fermats–Euler to the arithmetic of algebraic curves: about the history of diophantine equations after Euler

Talking about the Diophantine analysis’ history, namely, the problem of rational solutions of Diophantine equations, we should note the longevity of the algebraic approach, which goes back to Diophantus’ “Arithmetica”. Indeed, after the European mathematicians of the second half of the XVI century became acquainted with Diophantus’ oeuvre, algebraic apparatus of variable changes, substitutions and transformations turned into the main tool of finding rational solutions of Diophantine equations. Despite the limitations of this apparatus, there were obtained important results on rational solutions of quadratic, cubic and quartic indeterminate equations in two unknowns. Detailed historico-mathematical analysis of these results was done, inter alia, by I. G. Bashmakova and her pupils. The paper examines the departure from this algebraic treatment of Diophantine equations, typical for most of the research up to the end of XIX century, towards a more general viewpoint on this subject, characterized also by radical expansion of the tools used in the Diophantine equations’ investigations. The works of A. L. Cauchy, C. G. J. Jacobi and É. Lucas, where this more general approach was developed, are analyzed. Special attention is paid to the works of J. J. Sylvester on Diophantine equations and the paper “On the Theory of Rational Derivation on a Cubic Curve” by W. Story, which were not in the focus of the research on history of the Diophantine analysis and where apparatus of algebraic curves was used in a pioneering way.