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An improved estimate for certain Diophantine inequalities


Authors: Ming Chit Liu, Shu Ming Ng and Kai Man Tsang
Journal: Proc. Amer. Math. Soc. 78 (1980), 457-463
MSC: Primary 10B45; Secondary 10F05
DOI: https://doi.org/10.1090/S0002-9939-1980-0556611-3
MathSciNet review: 556611
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Abstract: Let $ {\lambda _1}, \ldots ,{\lambda _8}$ be any nonzero real numbers such that not all $ {\lambda _j}$ are of the same sign and not all ratios $ {\lambda _j}/{\lambda _k}$ are rational. If $ \eta ,\alpha $ are any real numbers with $ 0 < \alpha < 3/70$ then $ \vert\eta + \Sigma _{j = 1}^8{\lambda _j}n_j^3\vert < {(\max {n_j})^{ - \alpha }}$ has infinitely many solutions in positive integers $ {n_j}$.


References [Enhancements On Off] (What's this?)

  • [1] A. Baker, On some diophantine inequalities involving primes, J. Reine Angew. Math. 228 (1967), 166-181. MR 0217016 (36:111)
  • [2] R. J. Cook, Diophantine inequalities with mixed powers, J. Number Theory 9 (1977), 142-152. MR 0432540 (55:5528)
  • [3] I. Danicic, The solubility of certain diophantine inequalities, Proc. London Math. Soc. (3) 8 (1958), 161-176. MR 0096636 (20:3119)
  • [4] H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, J. London Math. Soc. 21 (1946), 185-193. MR 0020578 (8:565e)
  • [5] H. Davenport and K. F. Roth, The solubility of certain diophantine inequalities, Mathematika 2 (1955), 81-96. MR 0075989 (17:829e)
  • [6] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, New York and London, 1960.
  • [6a] K. W. Lau and M. C. Liu, Linear approximation by primes, Bull. Austral. Math. Soc. 19 (1978), 457-466. MR 536895 (80e:10032)
  • [7] M. C. Liu, Approximation by a sum of polynomials involving primes, J. Math. Soc. Japan 30 (1978), 395-412. MR 0485751 (58:5563)
  • [8] K. Ramachandra, On the sums $ \Sigma _{j = 1}^K{\lambda _j}{f_j}({p_j})$, J. Reine Angew. Math. 262/263 (1973), 158-165. MR 0327660 (48:6002)
  • [9] W. Schwarz, Über die Lösbarkeit gewisser Ungleichungen durch Primzahlen, J. Reine Angew. Math. 212 (1963), 150-157. MR 0191883 (33:110)
  • [10] R. C. Vaughan, Diophantine approximation by prime numbers. I, II, III, Proc. London Math. Soc. (3) 28 (1974), 373-384; ibid. (3) 28 (1974), 385-401; ibid. (3) 33 (1976), 177-192. MR 0337812 (49:2581)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0556611-3
Article copyright: © Copyright 1980 American Mathematical Society

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