Small prime solutions to linear equations in three variables

Tak Wing Ching¹ ; Kai Man Tsang¹

¹Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong

Acta Arithmetica, Tome 178 (2017) no. 1, pp. 57-76

Résumé

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 $b\gg A^{25}$.

Détail

DOI : 10.4064/aa8427-8-2016

Keywords: nonzero integers integer satisfying equiv pmod a suppose max obtain following improved bounds small prime solutions equation have sign there exist prime solutions satisfying max positive equation solvable