On the sum of a prime and two Fibonacci numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910986437394432 |
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| author | Xu, Ji-Zhen Chen, Yong-Gao |
| author_facet | Xu, Ji-Zhen Chen, Yong-Gao |
| contents | Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sum of a prime and two Fibonacci numbers Xu, Ji-Zhen Chen, Yong-Gao Number Theory 11P32, 11A41, 11B39, 11B13 Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities. |
| title | On the sum of a prime and two Fibonacci numbers |
| topic | Number Theory 11P32, 11A41, 11B39, 11B13 |
| url | https://arxiv.org/abs/2506.03631 |