Near-squares in binary recurrence sequences
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arXiv
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| Natura: | Preprint |
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2021
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| _version_ | 1866911747702521856 |
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| author | Tzanakis, Nikos Voutier, Paul |
| author_facet | Tzanakis, Nikos Voutier, Paul |
| contents | We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and $u_{n+2}(a)=au_{n+1}(a)-u_{n}(a)$ for $n \geq 0$. We show that for a given $a \geq 3$, there is at most one $n \geq 5$ such that $u_{n}(a)$ is a near-square. With the exceptions of $u_{6}(3)=12^{2}$ and $u_{7}(6)=239 \cdot 13^{2}$, any such $u_{n}(a)$ can only be a near-square if $a \equiv 2 \bmod 4$, $n \equiv 3 \bmod 4$ is prime and $n \geq 19$.
This is part of a more general phenomenon regarding near-squares in non-degenerate recurrence sequences defined for integers $a$ and $b=-b_{1}^{2}$ by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$ and $u_{n+2}(a,b)=au_{n+1}(a,b)+bu_{n}(a,b)$ for $n \geq 0$ (see our Conjecture 1.1). It arises from a new Aurifeuillean-like factorization of elements of recurrence sequences that we have discovered (see relation (1.1)). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_04523 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Near-squares in binary recurrence sequences Tzanakis, Nikos Voutier, Paul Number Theory 11B37, 11D61, 11R16, 11R04, 11R27, 11J86, 11Y40 We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and $u_{n+2}(a)=au_{n+1}(a)-u_{n}(a)$ for $n \geq 0$. We show that for a given $a \geq 3$, there is at most one $n \geq 5$ such that $u_{n}(a)$ is a near-square. With the exceptions of $u_{6}(3)=12^{2}$ and $u_{7}(6)=239 \cdot 13^{2}$, any such $u_{n}(a)$ can only be a near-square if $a \equiv 2 \bmod 4$, $n \equiv 3 \bmod 4$ is prime and $n \geq 19$. This is part of a more general phenomenon regarding near-squares in non-degenerate recurrence sequences defined for integers $a$ and $b=-b_{1}^{2}$ by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$ and $u_{n+2}(a,b)=au_{n+1}(a,b)+bu_{n}(a,b)$ for $n \geq 0$ (see our Conjecture 1.1). It arises from a new Aurifeuillean-like factorization of elements of recurrence sequences that we have discovered (see relation (1.1)). |
| title | Near-squares in binary recurrence sequences |
| topic | Number Theory 11B37, 11D61, 11R16, 11R04, 11R27, 11J86, 11Y40 |
| url | https://arxiv.org/abs/2106.04523 |