Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908792404312064 |
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| author | Baier, Stephan Bhandari, Arkaprava Haldar, Anup |
| author_facet | Baier, Stephan Bhandari, Arkaprava Haldar, Anup |
| contents | We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $λ_1x_1^2+\cdots +λ_nx_n^2\equiv λ_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $λ_1,...,λ_{n+1}$ are integer coefficients such that $(λ_1\cdots λ_{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $λ_1,...,λ_n>0$ and $(λ_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $λ_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $λ_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers Baier, Stephan Bhandari, Arkaprava Haldar, Anup Number Theory 11D09, 11D45, 11D79, 11E12, 11L05, 11L40 We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $λ_1x_1^2+\cdots +λ_nx_n^2\equiv λ_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $λ_1,...,λ_{n+1}$ are integer coefficients such that $(λ_1\cdots λ_{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $λ_1,...,λ_n>0$ and $(λ_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $λ_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $λ_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag. |
| title | Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers |
| topic | Number Theory 11D09, 11D45, 11D79, 11E12, 11L05, 11L40 |
| url | https://arxiv.org/abs/2406.12758 |