Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A lower bound for the two-variable Artin conjecture and prime divisors
  of recurrence sequences   In 1927, Artin conjectured that any integer other than -1 or a perfect square
generates the multiplicative group $\mathbb{Z}/p\mathbb{Z}^\times$ for
infinitely many $p$. In \cite{MoSt}, Moree and Stevenhagen considered a
two-variable version of this problem, and proved a positive density result
conditionally to the generalized Riemann Hypothesis by adapting a proof by
Hooley for the original conjecture (\cite{Ho}). In this article, we prove an
unconditional lower bound for this two-variable problem. In particular, we
prove an estimate for the number of distinct primes which divide one of the
first $N$ terms of a non-degenerate binary recurrence sequence. We also prove a
weaker version of the same theorem, and give three proofs that we consider to
be of independent interest. The first proof uses a transcendence result of
Stewart \cite{Stew}, the second uses a theorem of Bombieri and Schmidt on Thue
equations \cite{BoSc} and the third uses Mumford's gap principle for counting
points on curves by their height \cite{Mum}. We finally prove a disjunction
theorem, where we consider the set of primes satisfying either our two-variable
condition or the original condition of Artin's conjecture. We give an
unconditional lower bound for the number of such primes.

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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat