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Lengths of the intervals where integer polynomials can attain small values

https://doi.org/10.29235/1561-8323-2024-68-6-447-453

Abstract

The concept of the discriminant of a quadratic polynomial allows for easy extraction of information about its real and complex roots. The discriminant of a polynomial of an arbitrary degree is also an important characteristic of the polynomial, which proves useful in many problems in the theory of Diophantine approximation. In 2023, Belarusian mathematician D. Badziahin solved a problem posed by Davenport in the 1960s concerning the range of values of discriminants in the cubic case. The paper provides a complete solution to the problem of divisibility of discriminants by large powers of prime numbers in the case of cubic polynomials.

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ISSN 1561-8323 (Print)
ISSN 2524-2431 (Online)