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Polynomials which are near to k-th powers

Published online by Cambridge University Press:  24 October 2008

K. R. Matthews
K. R. Matthews
Affiliation:
Trinity College, Cambridge
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Extract

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

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 1 - 5
Copyright
Copyright © Cambridge Philosophical Society 1965

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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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