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

Harald A. Helfgott — 2007

Journal de Théorie des Nombres de Bordeaux

Let f [ x ] be a polynomial of degree d 3 without roots of multiplicity d or ( d - 1 ) . Erdős 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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