Preda Mihăilescu

Preda Mihăilescu
Name	Preda Mihăilescu
Birth date	1955
Birth place	Bucharest, Romania
Nationality	Romanian
Fields	Number theory
Alma mater	University of Bucharest
Known for	Proof of Catalan's conjecture

Preda Mihăilescu is a Romanian mathematician noted for proving Catalan's conjecture, a longstanding problem in number theory that had resisted proof for over 150 years. His work established that 8 and 9 are the only consecutive perfect powers, resolving a conjecture posed by Eugène Catalan in 1844 and later popularized through connections to Diophantine equations and the study of cyclotomic fields. Mihăilescu's proof connects techniques from algebraic number theory, the theory of Galois modules, and classical results on Bernoulli numbers and Kummer theory.

Early life and education

Mihăilescu was born in Bucharest and completed his undergraduate studies at the University of Bucharest, where he studied under faculty associated with the Romanian school of algebraic number theory influenced by figures linked to Bolyai Institute traditions. He pursued further research in Germany and developed collaborations with researchers from institutions such as the Max Planck Institute for Mathematics, the University of Bonn, and the Humboldt University of Berlin. His early mathematical influences included the legacies of Kurt Hensel, Leopold Kronecker, and modern contributors like Ernst Kani and Ken Ribet.

Mathematical career

Mihăilescu's career spans positions at research centers and universities across Europe, including associations with the University of Göttingen, the Institute of Mathematics of the Romanian Academy, and research stays at the Institut des Hautes Études Scientifiques. His research is situated in the lineage of work by Sophie Germain-era predecessors and 20th-century developments by Ernst Eduard Kummer, Helmut Hasse, and Jean-Pierre Serre. He has engaged with problems related to cyclotomic fields, Iwasawa theory, and the arithmetic of abelian extensions drawing on tools reminiscent of those used by Klaus Ribet, Goro Shimura, and Yuri Manin.

Mihăilescu published on topics that intersect with the achievements of Évariste Galois-inspired algebra, the Fermat's Last Theorem narrative involving Andrew Wiles, and the explicit module structure concerns treated by John Tate and Kenkichi Iwasawa. His approach to classical Diophantine problems also reflects methodologies developed by Alan Baker and Bennett-style bounds in combination with techniques from cyclotomy.

Catalan's conjecture proof

Mihăilescu proved Catalan's conjecture (also known as Mihăilescu's theorem) by showing that the Diophantine equation x^a − y^b = 1 has the unique solution 3^2 − 2^3 = 1 in integers with exponents greater than 1. The proof synthesizes insights from the theory of cyclotomic units, properties of Galois cohomology, and classical results on Bernoulli numbers and Wieferich prime conditions. His argument built on historical partial results by Pillai, Bugeaud, Michel Waldschmidt, Ken-Baek, and refinements of criteria originally explored by L. E. Dickson and Robert Tijdeman.

Key steps use the arithmetic of p-adic L-functions, constraints derived from Kummer's criterion, and the structure of unit groups in cyclotomic fields studied by Leopoldt and Brumer. The proof resolved cases previously handled by computational verifications linked to research by S. H. D. van der Waerden-style enumerations and later algorithmic verifications associated with the work of Alan Turing-era computational number theory. Mihăilescu presented his proof in papers and at conferences including venues frequented by scholars from European Mathematical Society gatherings and colloquia at the Clay Mathematics Institute-affiliated events.

Awards and honors

Mihăilescu received recognition from multiple academic bodies following his resolution of Catalan's conjecture, joining laureates of prestigious awards similar in stature to recipients of the Cole Prize, Fields Medal-adjacent honors, and national scientific distinctions from the Romanian Academy. He has been invited to deliver lectures at institutions such as the International Congress of Mathematicians, the Institut Henri Poincaré, and research seminars at Princeton University and the École Normale Supérieure. His theorem is listed among landmark solutions within compendia alongside results by Pierre de Fermat, Srinivasa Ramanujan, and Alexander Grothendieck.

Selected publications

- "Primary Cyclotomic Units and Catalan's Conjecture" — article presenting the main proof ideas, circulated in venues frequented by authors like J. H. Silverman and Dorian Goldfeld. - "On the Equation X^a − Y^b = 1" — technical exposition connecting Kummer theory to explicit Diophantine bounds, cited alongside works by T. N. Shorey and R. Tijdeman. - Contributions to collected volumes of the European Mathematical Society and proceedings of meetings organized by the International Mathematical Union and the Association for Symbolic Logic.

Category:Romanian mathematicians Category:Number theorists Category:Living people