# Questions tagged [diophantine-geometry]

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### Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$

Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer. While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
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### Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:  Solving genus zero diophantine ...
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### Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has $(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for $q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...