Abstract
Friedlander and Iwaniec investigated integral solutions \((x_1,x_2,x_3)\) of the equation \(x_1^2+x_2^2-x_3^2=D\), where D is square-free and satisfies the congruence condition \(D\equiv 5\bmod {8}\). They obtained an asymptotic formula for solutions with \(x_3\asymp M\), where M is much smaller than \(\sqrt{D}\). To be precise, their condition is \(M\ge D^{1/2-1/1332}\). Their analysis led them to averages of certain Weyl sums. The condition of D being square-free is essential in their work. We investigate the "opposite" case when \(D=n^2\) is a square of an odd integer n. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with \(x_3\asymp M\), where \(M\ge D^{1/2-1/16+\varepsilon }\).
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Acknowledgements
The author would like to thank the anonymous referee for valuable comments which helped to improve the exposition. He further would like to thank the Ramakrishna Mission Vivekananda Educational and Research Institute for providing excellent working conditions.
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Baier, S. Solutions of \(x_1^2+x_2^2-x_3^2=n^2\) with small \(x_3\). Ramanujan J 63, 293–337 (2024). https://doi.org/10.1007/s11139-023-00758-6
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DOI: https://doi.org/10.1007/s11139-023-00758-6