Faculty of Engineering and Natural Sciences
Jannis A. Antoniadis (Heraklion, Greece)
In 1844 Crelle's Journal published a note of Mr. E. Catalan where he stated the conjecture that the only (integral) solutions of the equation x^a- y^b = 1, where a > 1, b > 1, x > 0 and y > 0 are x = b = 3 and y = a = 2.
The conjecture has been proven to be true very recently by P. Mihailescu (2002). The whole history of the proof could be divided into three periods. During the first period we have some special results by using, more or less, elementary methods. The second period is connected with the theory of A.Baker about estimates for linear forms in logarithms. R. Tijdeman has proven that the number of solutions of a, b, x, y is finite and one can compute explicit bounds for the unknows. The upper bound is relatively small but still far beyond anything useful for practical purposes. At the end, the methods which have been used are of algebraic nature. Mihailescu uses algebraic number theory and, more specially, the theory of cyclotomic fields. Part of the whole proof was the result of Tijdeman but Yuri Bilu (2004) has given a proof "without logarithmic forms".
Wednesday, 3 May 2006 at 16:40, FENS 2019