Small prime solutions of a Diophantine equation with one prime and five cubes of primes

Abstract

Let $a_1,\cdots,a_6$ be non-zero integers satisfying $(a_i,a_j)=1, 1\leq i \lt j \leq 6$ and $b$ be any integer. For the Diophantine equation $a_1p_1+a_2p_2^3+\cdots+a_6p_6^3=b$ we prove that (i) if all $a_1,\cdots,a_6$ are positive and $b\gg \max \{|a_j|\}^{34+\varepsilon}$, then the equation is soluble in primes $p_j$, and (ii) if $a_1,\cdots,a_6$ are not all of the same sign, then the equation has prime solutions satisfying $\max \{ p_1,p_2^3,\cdots,p_6^3 \}\ll |b|+\max \{|a_j|\}^{33+\varepsilon}$, where the implied constants depend only on $\varepsilon$.