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Counting primitive integral solutions to spherical generalized Fermat equations
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 04 June 2026, pp. 1-17
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A solution $(x,y,z) \in \mathbb {Z}^3-\left \{ (0,0,0) \right \}$
to a generalized Fermat equation (1)$$ \begin{align} A\textsf{x}^{a} + B\textsf{y}^{b} + C\textsf{z}^{c} = 0 \end{align} $$is called primitive if $\gcd (x,y,z) = 1$
. By work of Beukers (1998, Duke Math. J., 91, 61–88), we know that in the spherical regime (i.e., when the Euler characteristic $\chi = \tfrac {1}{a} + \tfrac {1}{b} + \tfrac {1}{c} - 1$
is positive), if Equation (1) has one primitive solution, then it has infinitely many. In this work, we use the method of Fermat descent, as employed by Poonen et al. (2007, Duke Math. J., 137, 103–158), to refine Beukers’ result to an asymptotic count of the number of primitive integral solutions of bounded height.
