Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The rational part of a periodic continued fraction   Let $x$ be a periodic continued fraction with the initial block $0$ and the
repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form
$x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square.
The numbers $a$ and $\sqrt b$ are uniquely determined by $x$. In general it is
difficult to say what the influence of a certain digit of the repeating block
on the appearance of $x$ is. We highlight a noteworthy exception from this
rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit
$c_n$ of the repeating block, the fractional part of $2a$, however, is
independent of $c_n$. Of particular interest is the case $2a\in\mathbb Z$,
which occurs if, and only if, the sequence $c_1,\ldots,c_{n-1}$ is symmetric.

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The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: econ