Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 149–168

# 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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