Integers representable as a difference of two rational fourth powers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914509090717696 |
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| author | Ratcliffe, Ashleigh Xuan, Tho Nguyen |
| author_facet | Ratcliffe, Ashleigh Xuan, Tho Nguyen |
| contents | In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubility of the equation $n=x^4+y^4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$. Motivated by this, we investigate the equation $n=x^4-y^4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integers representable as a difference of two rational fourth powers Ratcliffe, Ashleigh Xuan, Tho Nguyen General Mathematics 14G05, 11D25, 11D41, 11Y50 In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubility of the equation $n=x^4+y^4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$. Motivated by this, we investigate the equation $n=x^4-y^4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$. |
| title | Integers representable as a difference of two rational fourth powers |
| topic | General Mathematics 14G05, 11D25, 11D41, 11Y50 |
| url | https://arxiv.org/abs/2604.15832 |