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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.12842 |
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| _version_ | 1866913400264589312 |
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| author | Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac Filipin, Alan Togbé, Alain |
| author_facet | Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac Filipin, Alan Togbé, Alain |
| contents | Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12842 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$ Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac Filipin, Alan Togbé, Alain Number Theory Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular. |
| title | On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2206.12842 |