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Note on the Diophantine equation \(1+x+x^ 2+\dots +x^ n=y^ m\). (English) Zbl 0703.11016

Elem. Math. 42, No. 3, 76 (1987).
Summary: The author considers some questions arising in connection with the lowest cases \(m=2\) and \(m=3\) of the title equation in view of W. Ljunggren’s theorem [Norsk Mat. Tidsskr. 25, 17–20 (1943; Zbl 0028.00901)] showing that the repunits (numbers of the form \(111\dots 11=1+10+\dots+10^n\)) are never squares or cubes for \(n\geq 1\).

MSC:

11D61 Exponential Diophantine equations
11A63 Radix representation; digital problems

Citations:

Zbl 0028.00901
Full Text: EuDML