On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915647808602112 |
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| author | Wu, Hai-Liang She, Yue-Feng |
| author_facet | Wu, Hai-Liang She, Yue-Feng |
| contents | In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge3$, we will show that for any sufficiently large integer $n$ with $\ord_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that
$$ \begin{cases}
x^2+y^2+z^2+w^2=n,
ax+by\in\mathcal{S},
\end{cases} $$
where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun's $x+24y$ conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture Wu, Hai-Liang She, Yue-Feng Number Theory In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge3$, we will show that for any sufficiently large integer $n$ with $\ord_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that $$ \begin{cases} x^2+y^2+z^2+w^2=n, ax+by\in\mathcal{S}, \end{cases} $$ where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun's $x+24y$ conjecture. |
| title | On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.23223 |