Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: For which 2-adic integers $x$ can $\sum_k \binom xk^{-1}$ be defined?   Let $f(n)=\sum_k \binom nk^{-1}$. In a previous paper, we defined for a
p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$,
where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd,
then -1 is the only element of $Z_p-N$ for which f(x) is p-definable. For p=2,
we proved that if the 1's in the binary expansion of x are eventually
extraordinarily sparse, then f(x) is 2-definable. Here we present some
conjectures that f(x) is 2-definable for many more 2-adic integers. We discuss
the extent to which we can prove these conjectures.

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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th