Small solutions of an equation with unequal powers of primes
Volume 195 / 2020
Acta Arithmetica 195 (2020), 57-68
MSC: 11P32, 11P55.
DOI: 10.4064/aa190107-8-8
Published online: 17 April 2020
Abstract
Let $a_1, \ldots , a_5$ be nonzero integers and $n$ any integer satisfying certain local conditions. Suppose also that $a_1, \ldots , a_5$ are pairwise coprime. We prove that if $a_j$ are not all of the same sign, then the equation $$ n=a_1p_1+a_2p_2^2+a_3p_3^3+a_4p_4^4+a_5p_5^5 $$ has prime solutions satisfying $\max \{p_1, p_2^2, p_3^3, p_4^4, p_5^5\} \ll |n|+(\max |a_j|)^{c+\varepsilon }$; and in parallel, if all $a_j$ are positive then the equation is soluble provided that $n \gg (\max |a_j|)^{c+1+\varepsilon }$, where $c=566/15=37.7333\ldots .$ Our method of treating the Waring–Goldbach problem for unequal powers of primes is more efficient and gives a better result than the conventional method.
