Exponential Diophantine equations

Abstract

The notion of variable, or unknown, appeared in the works of the Greek mathematician Diophantus, who lived (probably) during the third century a.d. He was particularly interested in the following question: does a given polynomial equation with integral (or rational) coefficients have a solution in integers (or in rational numbers)? Among the most classical examples is the equation x 2 + y 2 = z 2 , whose integral solutions give us the lengths of the sides of Pythagorean triangles. At that time (and, most probably, even since a few centuries before that time), all these solutions were perfectly known. Nowadays, we call Diophantine equation any polynomial equation with integer coefficients and whose unknowns are supposed to be rational integers. This definition is often extended to any type of equations involving integers and where the unknown are also integers. An emblematic example is Fermat's equation x n + y n = z n , where x, y, z and n > 3 are unknown positive integers. We often use the terminology “exponential Diophantine equation” when one or more exponents are unknown. The natural question is the following: an equation being given, determine the complete set of its integral solutions. Sometimes, this is quite easy, in particular when one can use congruences modulo a suitable integer. Let us for example consider the equation 3 m - 2 n = 1, which was solved by Levi ben Gershon (1288-1344), answering a question of the French composer Philippe de Vitry. Assume that there are integers m, n with n > 2 and 3m - 2n = 1. Then, 4 divides 3m - 1, whence m must be even. Writing m = 2k we obtain (3 k - 1) (3 k + 1) = 2n, which implies that both 3 k - 1 and 3 k + 1 are powers of 2. But the only powers of 2 which differ by 2 are 2 and 4. Hence k = 1 and we have proved that 3 2 - 2 3 = 1 is the only solution to 3 m - 2 n = 1 with n > 2. However, in most of the cases, to determine the complete set of integral solutions of a Diophantine equation remains an unsolved problem, and often it is even very difficult to prove whether this set is finite or not. When it is infinite, the next step is to give a complete description of all the integral solutions of the equation. For instance, the positive solutions of the equation 5x 2 - y 2 = ± 4 are precisely given by the integer pairs (Fn, Ln), where (Fn)n1 and (Ln)n1 are the Fibonacci and the Lucas sequences defined by F1 = F2 = 1, L1 = 1, L2 = 3, and satisfying Fn+2 = Fn+1 + Fn and Ln+2 = Ln+1 + Ln, for n > 1.