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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.22421 |
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| _version_ | 1866911111859666944 |
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| author | Karapetyan, Hayk |
| author_facet | Karapetyan, Hayk |
| contents | The polynomials $x^n + (1-x)^n + a^n$ arise naturally from FLT (Fermat's Last Theorem). We formulate a conjecture about them which is a generalization of FLT. We investigate the complex roots of these polynomials, and our main result is that in the cases $|a|\leq \frac12$ and $a=-1$, they lie on an explicitly given curve while 'filling in' that curve. We hypothesize that this property can be generalized to hold for other $a$ as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a property of the polynomials $x^n + (1-x)^n + a^n$ Karapetyan, Hayk Number Theory The polynomials $x^n + (1-x)^n + a^n$ arise naturally from FLT (Fermat's Last Theorem). We formulate a conjecture about them which is a generalization of FLT. We investigate the complex roots of these polynomials, and our main result is that in the cases $|a|\leq \frac12$ and $a=-1$, they lie on an explicitly given curve while 'filling in' that curve. We hypothesize that this property can be generalized to hold for other $a$ as well. |
| title | On a property of the polynomials $x^n + (1-x)^n + a^n$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.22421 |