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A note on the diophantine equation \({x^ m -1 \over x-1} = y^ n\). (English) Zbl 0783.11013

Let \(\mathbb{P}\) be the set of primes and prime powers. The following result is obtained: if the solution \((x,y,m,n)\) in integers \(x>1\), \(y>1\), \(m>2\), \(n>1\) of the equation \[ {{x^ m-1} \over {x-1}}= y^ n \] satisfies \(x\in\mathbb{P}\) and \(y\equiv 1\pmod x\), then \(x^ m<C\) for an effectively computable absolute constant \(C\). In fact the author needs to prove this only for odd integers \(m\) and \(n\) as for even \(n\) all solutions are explicitly known [W. Ljunggren, Norsk Mat. Tidsskr. 25, 17-20 (1943; Zbl 0028.00901)] and for even \(m\) T. Shorey and R. Tijdeman proved the corresponding result in [Math. Scand. 39, 5-18 (1976; Zbl 0341.10017)].

MSC:

11D61 Exponential Diophantine equations
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