Diophantus of Alexandria

Works and surviving material

Diophantus' best-known work is the Arithmetica, a multi-volume treatise that presented numerical problems and methods of solution. Parts of the original work survive in Greek; additional material circulated in later translations. Because only a portion of his original writings remains, our picture of his full output is reconstructed from surviving books, commentaries, and medieval translations.

Methods and mathematical style

Rather than modern symbolic notation, Diophantus used a compact syncopated form of algebra: abbreviations and conventions to represent unknowns and powers, with explanations in words interwoven. He solved linear and quadratic problems and many indeterminate problems that admit infinitely many solutions. Problems treated by Diophantus include finding integers or rational numbers satisfying several conditions simultaneously — the class of problems now called Diophantine equations.

Typical problems and techniques

• Solving polynomial equations in one unknown by transformation to simpler forms.
• Constructing rational solutions to systems by parameterization.
• Handling sums, products, and relationships among numbers with ad hoc but systematic manipulations.

Influence and later developments

Diophantus' work had a long afterlife. His problem-centered style influenced Islamic mathematicians and, through translations, European scholars in the Renaissance. A marginal note made by Pierre de Fermat on one problem of the Arithmetica famously claimed there was no nontrivial integer solution for a certain exponent — a remark that motivated centuries of work and ultimately the proof of Fermat's Last Theorem. Modern number theory studies the general class of questions he explored; this field is often described as the study of integer and rational solutions to polynomial equations.

Notable facts

Diophantus is often called a founder or early pioneer of algebraic problem solving. His legacy also includes a well-known epitaph riddle recounting the stages of his life, which has been reproduced and discussed in later literature. For general background and biographical summaries see sources listed under classical biographies and mathematical histories.