Power-free values, large deviations, and integer points on irrational curves

Details Power-free values, large deviations, and integer points on irrational curves Power-free values, large deviations, and integer points on irrational curves

2004-11-16

arxiv.org/abs/math/0411369v3

Mathematics

Number Theory

39 pages; rather major revision, with strengthened and generalized statements

Scientific paper

Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erd\H{o}s conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.

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