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Polynomials which are near to k-th powers
Published online by Cambridge University Press: 24 October 2008
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1. Let f(x) be a polynomial of degree n ≥ 2 with integral coefficients, the highest coefficient being positive. It is well known that if f(x) is an exact k-th power for all sufficiently large integers x, where k ≥ 2, then f(x) = g(x)k identically, where g(x) is another polynomial with integral coefficients. (See Pólya and Szegö (4), section 8, problems 114, 190; also Davenport, Lewis and Schinzel(1), where other references are given.) The main purpose of this note is to prove that if we suppose only that
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- Research Article
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- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 1 - 5
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- Copyright © Cambridge Philosophical Society 1965
References
REFERENCES
(1)Davenport, H., Lewis, D. J. and Schinzel, A.Polynomials of certain special types. Acta Arith. 9 (1964), 107–116.CrossRefGoogle Scholar
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(4)Pólya, G. and Szegö, G.Aufgaben und Lehrsätze aus der Analysis, vol. ii (Berlin, 1925).Google Scholar
(6)Weyl, H.Ueber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
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