Schoof - Catalans Conjecture

Ren Schoof 1

(1)

Abstract

In this book, we present Preda Mihilescus beautiful proof of the conjecture made by Eugne Charles Catalan in 1844 in a letter [11] to the editor of Crelles journal.

In this book, we present Preda Mihilescus beautiful proof of the conjecture made by Eugne Charles Catalan in 1844 in a letter [11] to the editor of Crelles journal:

In other words, Catalan proposed the following.

When is fixed, the k th powers of natural numbers are necessarily far apart. However, when one varies k , two powers can be closer to one another than one might expect. For instance, we have and . Catalan conjectured that the only powers for which the difference is as small as 1 are 32 and 23.

Phrased in yet another way, Catalan conjectured that for exponents , the Diophantine equation admits no solution in natural numbers other than the one given by x =3, p =2 and y =2, .

Apparently, Catalan himself did not get very far in solving the problem. We read this in a note [12] that was published more than forty years after his 1844 letter. Here Catalan reports on his early attempts:

The Socit Belge des Professeurs de Mathmatique dExpression Franaise has published a book on the academic and political activities of Eugne Catalan [20]. It contains a reproduction of a painting of Catalan which is at present in the possession of the Universit de Lige (Fig. ).

In this book, we mainly concentrate on Preda Mihilescus proof of Catalans famous conjecture. We discuss earlier work only when it is relevant to our presentation of the proof. For an overview of earlier work on the conjecture, see [7, 14, 15, 35, 38]. We only mention one important result: In 1976, Rob Tijdeman [49] (Fig. ) showed that there exist only finitely many pairs of consecutive perfect powers. His proof is based on the theory of linear forms in logarithms. Unfortunately, the bound on the size of the solutions that came out of Tijdemans proof is astronomical. There remained a large gap between the relatively small exponents p,q for which Catalans equation had been solved and Tijdemans estimates. In the successive years, this gap was narrowed considerably by various people [2, 6, 17, 19, 26, 32, 33, 34, 44, 46]. But the gap remained very large. In 1999, refinements of Tijdemans estimates were shown to imply Catalans conjecture when both exponents p,q exceed . On the other hand, elaborate computer calculations had proven the conjecture when one of p,q is smaller than 105. See [34] for more information.

Between 2000 and 2003, Preda Mihilescu (Fig. ) proved three theorems concerning Catalans conjecture:

Mihilescus proofs of Theorems I, II, and III appear in [35], [36], and [37], respectively. We show now that these three theorems together lead to a proof of Catalans conjecture. This proof does not rely on Tijdemans work or on the computer calculations mentioned above.