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arXiv · 2503.22421

On a property of the polynomials $x^n + (1-x)^n + a^n$

Hayk Karapetyan
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Abstract

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.

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BibTeXRIS

Hayk Karapetyan. 2025-03-28. On a property of the polynomials $x^n + (1-x)^n + a^n$. https://arxiv.org/abs/2503.22421

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