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Power free values of polynomials

Published online by Cambridge University Press:  26 February 2010

M. Nair
M. Nair
Affiliation:
Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD.
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If an integer does not have a k-th power of a positive integer, other than 1, for a divisor, it is said to be k–free. Let f(n) be an irreducible polynomial, with rational integer coefficients, of degree g, having no fixed k-th power divisors other than 1. We define

i.e. Nk(x) is the number of positive integers n not exceeding x such that f(n) is k-free. One would expect that f(n) is square-free for infinitely many n and further that, given x sufficiently large, there is an n with x < n ≤ x + h, such that f(n) is square-free for h = 0(x2) where ε is any real number > 0. These conjectures, however, seem to be extraordinarily difficult to prove. We begin with a brief account of the best results that have been attained so far.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 2 , December 1976 , pp. 159 - 183
Copyright
Copyright © University College London 1976

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References

1.Roth, K. F.. “On the gaps between square-free numbers”, J. London Math. Soc, 26 (1951), 263–268.CrossRefGoogle Scholar
2.Halberstam, H. and Roth, K. F.. “On the gaps between consecutive A–free numbers”, J. London Math. Soc, 26 (1951), 268–273.CrossRefGoogle Scholar
3.Hooley, C.. “On the power free values of polynomials”, Mathematika, 14 (1967), 21–26.CrossRefGoogle Scholar
4.Nagell, T.. “Zur Arithmetik der Polynome”, Abhandl. Math. Sem. Hamburg, 1 (1922), 179–184.Google Scholar
5.Ricci, G.. “Ricerche aritmetiche sui polynomi”, Rend. Circ. Mat. Palermo, 57 (1933), 433–475.CrossRefGoogle Scholar
6.Erdős, P.. “Arithmetical properties of polynomials”, J. London Math. Soc, 28 (1953), 416–425.CrossRefGoogle Scholar
7.Cugiani, M.. “Sugli intervalli frai valori deirargomento pei quali un polinomio risulta libero da potenze”, Riv. Mat. Univ. Parma, 4 (1953), 95–103.Google Scholar
8.Tijdeman, R.. “On the maximal distance of numbers with a large prime factor”, J. London Math. Soc. (2), 5 (1972), 313–320.CrossRefGoogle Scholar
9.Baker, A.. Transcendental Number Theory (Cambridge, University Press, 1975).CrossRefGoogle Scholar
10.Parry, C. J.. “The p-adic generalisation of the Thue-Siegel Theorem”, Ada Mathematica, 83 (1950), 1–100.CrossRefGoogle Scholar
11.Sukthankar, N. S.. “On Grimm's conjecture in algebraic number fields”, Indagationes Mathematicae, 35.Google Scholar
12.Le Veque, W. J.. Topics in Number Theory, Vol. II (Addison-Wesley, 1961).Google Scholar
13.Huxley, M. N.. The Distribution of Prime Numbers, Oxford Mathematical Monographs (Clarendon Press, Oxford, 19 ).Google Scholar
14.Huxley, M. N. and Iwaniec, H.. “Bombieri's Theorem in short intervals”, Mathematika, 22 (1975), 188–194.CrossRefGoogle Scholar
15.Richert, H.-E.. “Selberg's sieve with weights”, Mathematika, 16 (1969), 1–22.CrossRefGoogle Scholar
16.Hooley, C.. Applications of sieve methods to the theory of numbers (CambridgeUniversity Press, (1976).Google Scholar