Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of Sondow's conjecture on the Smarandache function   The Smarandache function of a positive integer $n$, denoted by $S(n)$, is
defined to be the smallest positive integer $j$ such that $n$ divides the
factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the
inequality $n^k < S(n)!$ holds for almost all positive integers $n$. This
confirms Sondow's conjecture which asserts that the inequality $n^2 < S(n)!$
holds for almost all positive integers $n$.

24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin