359 research outputs found
Power-free values, large deviations, and integer points on irrational curves
Let be a polynomial of degree
without roots of multiplicity or . Erd\H{o}s conjectured that, if
satisfies the necessary local conditions, then is free of th
powers for infinitely many primes . This is proved here for all 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.Comment: 39 pages; rather major revision, with strengthened and generalized
statement
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