on ”NUMBER THEORY AND MATHEMATICAL PHYSICS” On recent Diophantine results

Diophantus of Alexandria was a greek mathematician, around 200 AD, who studied mathematical problems, mostly geometrical ones, which he reduced to equations in rational integers or rational numbers. He was interested in producing at least one solution. Such equations are now called Diophantine equations. An example is y − x = 1, a solution of which is (x = 2, y = 3). More generally, a Diophantine equation is an equation f(x1, . . . , xn) = 0, where f ∈ Z[X1, . . . , Xn] is a given polynomial with rational integer coefficients, while the unknowns x1, . . . , xn are either rational integers or rational numbers. Hilbert’s tenth problem is to give an algorithm answering the question of whether such an equation has a solution or not (it is known since 1970 that there is no such algorithm for integral solutions — see [12]). One speaks of an exponential Diophantine equation when some of the exponents are unknown. A well known example is Fermat’s equation x + y = z, where the unknowns are the positive rational integers x, y, z, n and n ≥ 2 (see again [12]). Other examples are Catalan’s equation x − y = 1, where the unknowns are the rational integers (x, y, p, q) with p and q both ≥ 2 and the more general Pillai’s equation x − y = k, where k ≥ 1 is fixed and the unknowns are again (x, y, p, q). After Diophantus who was interested finding at least one solution, Pierre de Fermat considered the question of finding all solutions. Nowadays one of the most efficient tool for solving Diophantine equations is Diophantine approximation theory, which studies the approximation of real or complex