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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.

Prime constellations in triangles with binomial coefficient congruences

Larry Ericksen (2009)

Acta Mathematica Universitatis Ostraviensis

The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by...

Prime factors of values of polynomials

J. Browkin, A. Schinzel (2011)

Colloquium Mathematicae

We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.

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