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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 $f \in ℤ [x]$ be a polynomial of degree $d \geq 3$ 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.

Enumeration of tilings of diamonds and hexagons with defects.

Helfgott, Harald A.; Gessel, Ira M. — 1999

The Electronic Journal of Combinatorics [electronic only]

Page 1

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