New results similar to Lagrange's four-square theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917857759068160 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$.
Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New results similar to Lagrange's four-square theorem Sun, Zhi-Wei Number Theory 11B13, 11E20, 11E25, 11D85, 11P05 In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers. |
| title | New results similar to Lagrange's four-square theorem |
| topic | Number Theory 11B13, 11E20, 11E25, 11D85, 11P05 |
| url | https://arxiv.org/abs/2411.14308 |