On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$
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arXiv
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| Format: | Preprint |
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2018
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| author | Wu, Hai-Liang Ni, He-Xia Pan, Hao |
| author_facet | Wu, Hai-Liang Ni, He-Xia Pan, Hao |
| contents | In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$.
In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_10136 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ Wu, Hai-Liang Ni, He-Xia Pan, Hao Number Theory In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$. |
| title | On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/1806.10136 |