On the Diophantine equation \({x^ n-1\over x-1}=y^ q\).
The paper deals with the Diophantine equation \[ \frac{x^n-1}{x-1}=y^q\;\text{ in \;integers}\;x>1,\;y>1,\;n>2,\;q\geq 2.\tag{1} \] It has been conjectured that this equation has only the solutions \[ \frac{3^5-1}{3-1}=11^2, \quad \frac{7^4-1}{7-1}=20^2\quad \text{and}\quad \frac{18^3-1}{18-1}=7^3. \] This conjecture is still open. Ljunggren solved it for \(q=2\) and Nagell when \(3| n\) and \(4| n.\) The equation is completely solved when \(x\) is a square. This latter result depends on several methods in Diophantine approximation together with heavy computer calculations. These methods and long computations were used in this paper to show that the equation (1) implies that there exists a prime divisor of \(x\) which does not divide \(y-1.\) It is known that every solution of (1) with \(y\) an odd prime satisfies \(n=\text{ord}_y x.\) Using the above result the authors were able to show that \(q=\text{ord}_x y\) answering a question of Edgar. Irrationality of certain numbers is also obtained.
scientific article; zbMATH DE number 1537814