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Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 149–168 | Cite as

The exceptional set for Diophantine inequality with unlike powers of prime variables

  • Wenxu Ge
  • Feng Zhao
Article

Abstract

Suppose that λ1, λ2, λ3, λ4 are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ12 is algebraic and irrational. Denote by E(V,N, δ) the number of vV with vN such that the inequality
$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$
has no solution in primes p1, p2, p3, p4. We show that
$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$
for any ɛ > 0.

Keywords

Davenport-Heilbronn method prime varaible exceptional set Diophantine inequality 

MSC 2010

11D75 11P32 11P55 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Information SciencesNorth China University of Water Resources and Electric PowerHenanP. R. China

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