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Linear approximation by primes

Published online by Cambridge University Press:  17 April 2009

Kee-Wai Lau  and
Ming-Chit Liu
Kee-Wai Lau
Affiliation:
Department of Mathematics, University of Hong Kong, Hong Kong.
Ming-Chit Liu
Affiliation:
Department of Mathematics, University of Hong Kong, Hong Kong.
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Abstract

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In this present paper we shall prove the following. Suppose that λ1, λ2, λ3 are any non-zero real numbers not all of the same sign and that λ1/λ2 is irrational. If η is any real number and, 0 < α < 1/9, then there are infinitely many prime triples (p1, p2, p3) for which

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 19 , Issue 3 , December 1978 , pp. 457 - 466
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Baker, A., “On some diophantine inequalities involving primes”, J. reine angew. Math. 228 (1967), 166–181.Google Scholar
[2]Cook, R.J., “Diophantine inequalities with mixed powers”, J. Number Theory 9 (1977), 142–152.CrossRefGoogle Scholar
[3]Davenport, H. and Heilbronn, H., “On indefinite quadratic forms in five variables”, J. London Math. Soc. 21 (1946), 185–193.CrossRefGoogle Scholar
[4]Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, 4th edition (Clarendon Press, Oxford, 1960).Google Scholar
[5]Liu, Ming-Chit, “Approximation by a sum of polynomials involving primes”, J. Math. Soc. Japan 30 (1978), 395–412.CrossRefGoogle Scholar
[6]Montgomery, Hugh L., Topics in multiplicative number theory (Lecture Notes in Mathematics, 227. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[7]Ramachandra, K., “On the sums ”, J. reine angew. Math. 262/263 (1973), 158–165.Google Scholar
[8]Schwarz, Wolfgang, “Über die Lösbarkeit gewisser Ungleichungen durch Primzahlen”, J. reine angew. Math. 212 (1963), 150–157.CrossRefGoogle Scholar
[9]Vaughan, R.C., “Diophantine approximation by prime numbers, I”, Proc. London Math. Soc. (3) 28 (1974), 373–384.CrossRefGoogle Scholar
[10]Vaughan, R.C., “Diophantine approximation by prime numbers, II”, Proc. London Math. Soc. (3) 28 (1974), 385–401.CrossRefGoogle Scholar
[11]Vaughan, R.C., “Diophantine approximation by prime numbers, III”, Proc. London Math. Soc. (3) 33 (1976), 177–192.CrossRefGoogle Scholar