On the Diophantine equation (xⁿ - 1)/(x-1)=y^q
by
Yann Bugeaud
Université Louis Pasteur I.R.M.A. 7, rue Descartes 67084 STRASBOURG (FRANCE)

We present a survey of recent results on the Diophantine equation (E) : (xⁿ - 1)/(x-1) = y^q, in integers x > 1, n > 2, y > 1, q > 1, which was studied by (among others) Ljunggren, Nagell and Inkeri. These new statements have been proved in several works by Bennett, Bugeaud, Mignotte, Roy, Saradha, Shorey, and they include the following ones :

- (E) has no solution with x being a square;

- (E) has no solution (x, y, n, q) with n congruent to 1 modulo q, except (3, 11, 5, 2);

- if (E) has a solution (x, y, n, q) distinct from (18, 7, 3, 3), then there exists a prime p such that p divides x and q divides p-1.

In particular, with Mignotte, we have proved that an integer > 1 with all digits equal to 1 in base ten cannot be a pure power.

Date received: February 3, 1999