Ratios of two powers of van der Laan-Padovan numbers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917006563868672 |
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| author | Yamada, Tomohiro |
| author_facet | Yamada, Tomohiro |
| contents | The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0, 1, \ldots$. We determine all pairs $(P_m, P_n)$ satisfying $P_m^b=2^{g_1} 3^{g_2} 5^{g_3} 7^{g_4} P_n^a$ for some integers $g_1, g_2, g_3, g_4$, $a$, and $b$. More generally, for a linear recurrence sequence $u_n$ satisfying the dominant root condition and a given set of primes $p_1, \ldots, p_k$, there exist only finitely many pairs $(u_m, u_n)$ satisfying $u_m^b=p_1^{g_1} \cdots p_k^{g_k} u_n^a$ for some integers $g_1, \ldots, g_k$, $a$, and $b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ratios of two powers of van der Laan-Padovan numbers Yamada, Tomohiro Number Theory 11A05, 11B37, 11B83, 11D61, 11J86 The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0, 1, \ldots$. We determine all pairs $(P_m, P_n)$ satisfying $P_m^b=2^{g_1} 3^{g_2} 5^{g_3} 7^{g_4} P_n^a$ for some integers $g_1, g_2, g_3, g_4$, $a$, and $b$. More generally, for a linear recurrence sequence $u_n$ satisfying the dominant root condition and a given set of primes $p_1, \ldots, p_k$, there exist only finitely many pairs $(u_m, u_n)$ satisfying $u_m^b=p_1^{g_1} \cdots p_k^{g_k} u_n^a$ for some integers $g_1, \ldots, g_k$, $a$, and $b$. |
| title | Ratios of two powers of van der Laan-Padovan numbers |
| topic | Number Theory 11A05, 11B37, 11B83, 11D61, 11J86 |
| url | https://arxiv.org/abs/2510.06192 |