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On the Sum

Published online by Cambridge University Press:  20 January 2009

Sean Mcdonagh
Sean Mcdonagh
Affiliation:
University College Galway
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Erdős (1) has proved the following result:

Theorem A.Every integral polynomial g(n) of degree k ≧ 3, represents for infinitely many integers n a(k-1)th power-free integer provided, in the case where k is a power of 2, there exists an integer n such that g(n)≢0 (mod 2k-1).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 3 , June 1967 , pp. 215 - 219
Copyright
Copyright © Edinburgh Mathematical Society 1967

References

REFERENCES

(1) Erdös, P., Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), 416–425.CrossRefGoogle Scholar
(2) Erdös, P., On the sum J. London Math. Soc. 27 (1952), 7–15.CrossRefGoogle Scholar
(3) Van Der Corput, J. G., Une inégalité relative au nombres des diviseurs, Proc. Ron. Ned. Akad. Wet. Amsterdam, 42 (1939), 547–553.Google Scholar
(4) Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 4th ed. (Oxford, 1960).Google Scholar
(5) Hooley, C., On the representation of a number as the sum of a square and a product, Math. Zeitschrift, 69 (1958), 211–227.CrossRefGoogle Scholar