Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-28T18:03:59.080Z Has data issue: false hasContentIssue false

Perfect powers in values of certain polynomials at integer points

Published online by Cambridge University Press:  24 October 2008

T. N. Shorey
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Bombay 400005, India

Extract

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d1, …, dt with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equation

Put

so that ½ < vt ≤ ¾. If α > 1 and kα < mkl, then equation (1) implies that

for 1 ≤ it and hence

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baker, A.. Rational approximations to 3√2 and other algebraic numbers. Quart. J. Math. Oxford (2) 15 (1964), 375383.CrossRefGoogle Scholar
[2]Baker, A.. Contributions to the theory of Diophantine equations I: On the representation of integers by binary forms. Philos. Trans. Roy. Soc. London Ser. A 263 (1968), 173208.Google Scholar
[3]Baker, A.. Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc. 65 (1969), 439444.CrossRefGoogle Scholar
[4]Baker, A.. The theory of linear forms in logarithms. In Transcendence Theory: Advances and Applications (Academic Press, 1977), 127.Google Scholar
[5]ErdÖs, P.. Note on the product of consecutive integers (II). J. London Math. Soc. 14 (1939), 245249.CrossRefGoogle Scholar
[6]Erdös, P.. On a diophantine equation. J. London Math. Soc. 26 (1951), 176–178.CrossRefGoogle Scholar
[7]Erdös, P.. On the product of consecutive integers III. Indag. Math. 17 (1955), 8590.CrossRefGoogle Scholar
[8]Erdös, P. and Selfridge, J. L.. The product of consecutive integers is never a power. Illinois J. Math. 19 (1975), 292301.CrossRefGoogle Scholar
[9]Erdös, P. and Turk, J.. Products of integers in short intervals. Acta Arith. 44 (1984), 147174.CrossRefGoogle Scholar
[10]Inkeri, K.. On the diophantine equation a(x n – 1)/(x – 1) = y m. Acta Arith. 21 (1972), 299311.CrossRefGoogle Scholar
[11]Ljunggren, W.. Some theorems on indeterminate equations of the forms (x n – 1)/(x – 1) = y q [Norwegian]. Norsk. Mat. Tidsskr. 25 (1943), 1720.Google Scholar
[12]Nagell, T.. Note sur l' équation indéterminée (x n – 1)/(x – 1) = y q. Norsk. Mat. Tidsskr. (1920), 7578.Google Scholar
[13]Nagell, T.. Sur l' équation indéterminée (x n – 1)/(x – 1) = y 2. Norsk. Mat. Forenings Skrifter 1 (1921), no. 3, 17 pp.Google Scholar
[14]Shorey, T. N.. Linear forms in the logarithms of algebraic numbers with small coefficients I. J. Indian Math. Soc. 38 (1974), 271284.Google Scholar
[15]Shorey, T. N.. On gaps between numbers with a large prime factor II. Acta Arith. 25 (1974), 365373.CrossRefGoogle Scholar
[16]Shorey, T. N. and Tijdeman, R.. New applications of Diophantine approximations to Diophantine equations. Math. Scand. 39 (1976), 518.CrossRefGoogle Scholar
[17]Shorey, T. N., van der Poorten, A. J., Tijdeman, R. and Schinzel, A.. Applications of Gel'fond-Baker method to Diophantine equations. In Transcendence Theory: Advances and Applications (Academic Press, 1977), 5977.Google Scholar
[18]Sprindžuk, V. G.. Hyperelliptic diophantine equations and class numbers of ideals [Russian] Acta Arith. 30 (1976), 95108.Google Scholar
[19]Waldschmidt, M.. A lower bound for linear forms in logarithms. Acta Arith. 37 (1980), 257283.CrossRefGoogle Scholar