The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911393896202240 |
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| author | Novaković, Saša |
| author_facet | Novaković, Saša |
| contents | Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for equations of the form $P(x)=\prod_{i=1}^rH_{n_i}$, where $P(x)$ is a polynomial and $H_{n_i}$ are divisible sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16757 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$ Novaković, Saša Number Theory 11D61, 11D72, 11D59, 14G05 Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for equations of the form $P(x)=\prod_{i=1}^rH_{n_i}$, where $P(x)$ is a polynomial and $H_{n_i}$ are divisible sequences. |
| title | The Diophantine equation $P(x)=\overset{r}{\underset{i=1}{\prod}}H_{n_i}$ |
| topic | Number Theory 11D61, 11D72, 11D59, 14G05 |
| url | https://arxiv.org/abs/2601.16757 |