PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Small prime solutions to linear equations in three variables

Volume 178 / 2017

Tak Wing Ching, Kai Man Tsang 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 $b\gg A^{25}$.

Authors

  • Tak Wing ChingDepartment of Mathematics
    The University of Hong Kong
    Pokfulam Road, Hong Kong
    e-mail
  • Kai Man TsangDepartment of Mathematics
    The University of Hong Kong
    Pokfulam Road, Hong Kong
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image