Bera, Ranjan and Banerjee, Pradipto
(2020)
On the nearest irreducible lacunary neighbour to an integer polynomial.
Colloquium Mathematicum, 162 (1).
pp. 121134.
ISSN 00101354
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Abstract
There is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with λ ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán.
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