ON THE DIOPHANTINE EQUATION x m − 1 x − 1 = yn − 1 y − 1
نویسندگان
چکیده
There is no restriction in assuming that y > x in (1) and thus we have m > n. This equation asks for integers having all their digits equal to one with respect to two distinct bases and we still do not know whether or not it has finitely many solutions. Even if we fix one of the four variables, it remains an open question to prove that (1) has finitely many solutions. However, when either the bases x and y, or the base x and the exponent n, or the exponents m and n are fixed, then it is proved that (1) has finitely many solutions (see [3] for references). In the first two cases, thanks to Baker’s theory of linear forms in logarithms, we can compute explicit (huge) upper bounds for the size of the solutions. As for the number of solutions, Shorey [14] proved that for two integers y > x, the Diophantine equation
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