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Exercise $12$ in Section $1.6$ of Nathanson's : Methods in Number Theory book has the following question.

  • When is the sum of a geometric progression equal to a power? Equivalently, what are the solutions of the exponential diophantine equation $$1+x+x^{2}+ \cdots +x^{m} = y^{n} \qquad \cdots \ (1)$$ in integers $x,m,n,y$ greater than $2$? Check that \begin{align*} 1 + 3 + 3^{2} + 3^{3} + 3^{4} & = 11^{2}, \\\ 1 + 7 + 7^{2} + 7^{3} &= 20^{2}, \\\ 1 + 18 +18^{2} &= 7^{3}. \end{align*} These are the only known solutions of $(1)$.

The Wikipedia link doesn't reveal much about the above question. My question here would be to ask the following:

  • Are there any other known solutions to the above equation. Can we conjecture that this equation can have only finitely many solutions?

Added: Alright. I had posted this question on Mathoverflow some time after I had posed here. This user by name Gjergji Zaimi had actually given me a link which tells more about this particular question. Here is the link:

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    Calculate the genus and you'll have your conjecture on the finitude of points.2012-06-01
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    @ex0du5 I apologize, I am not well versed in Algebraic Geometry2012-06-01
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    This problem is discussed in detail in the book *Catalan's conjecture: are 8 and 9 the only consecutive powers?* by Paulo Ribenboim.2012-06-01
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    @Chandrasekhar : Really nice question man..+1.2012-06-01
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    @Iyengar: Thanks. Not my question really. I should thank Nathanson, and perhaps the Greeks who were so much interested in Diophantine equations.2012-06-01
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    @Chandrasekhar : Ya, We are but dwarfs mounted on the shoulders of giants, so that we can see more and further than they; yet not by virtue of the keenness of our eyesight, nor through the tallness of our stature, but because we are raised and borne aloft upon that giant mass.2012-06-01
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    +1 at least this diophantine equation has some motivation.2012-06-01

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