Small prime solutions to linear equations in three variables
Volume 178 / 2017
Acta Arithmetica 178 (2017), 57-76
MSC: Primary 11P32; Secondary 11P55.
DOI: 10.4064/aa8427-8-2016
Published online: 8 February 2017
Abstract
Let $a_1,a_2,a_3$ be nonzero integers and $b$ be any integer satisfying $b\equiv a_1+a_2+a_3\pmod{2}$ and $(b,a_i,a_j)=1$ for $1\le i \lt j\le 3$. Suppose $(a_1,a_2,a_3)=1$ and $A=\max{\{| a_1|,| a_2|,| a_3|\}}$. We obtain the following improved bounds for small prime solutions of the equation $a_1p_1+a_2p_2+a_3p_3=b$:
(i) if not all of $a_1,a_2,a_3$ have the same sign, then there exist prime solutions satisfying $\max_{1\le j\le 3}| a_j| p_j\ll| b|+A^{25}$;
(ii) if $a_1,a_2,a_3$ are all positive, then the equation $a_1p_1+a_2p_2+a_3p_3=b$ is solvable for $b\gg A^{25}$.
