Polynomial $D(4)$-quadruples over Gaussian Integers
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913400303386624 |
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| author | Trebješanin, Marija Bliznac Babić, Sanda Bujačić |
| author_facet | Trebješanin, Marija Bliznac Babić, Sanda Bujačić |
| contents | A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\mathbb{Z}[i][X]$.
In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_10575 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polynomial $D(4)$-quadruples over Gaussian Integers Trebješanin, Marija Bliznac Babić, Sanda Bujačić Number Theory 11D09, 11D45 A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\mathbb{Z}[i][X]$. In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$. |
| title | Polynomial $D(4)$-quadruples over Gaussian Integers |
| topic | Number Theory 11D09, 11D45 |
| url | https://arxiv.org/abs/2210.10575 |