An improved bound for prime solutions of some ternary equations
type: Item
instance of: scholarly article
title: An improved bound for prime solutions of some ternary equations
zbMATH Open document ID: 0602.10040
DOI: 10.1007/BF01161922
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173855
review text: Let \(b_ 1,b_ 2,b_ 3\) be non-zero integers, not all of the same sign, such that \((b_ j,b_ k)=1\) for \(j\neq k\). Let n be either 1 or 2 satisfying \(\sum ^{3}_{j=1}b_ j\equiv n (mod 2)\). In this paper we prove that there are positive absolute constants A and \(B_ 0\) such that the equation \[ b_ 1 p_ 1+b_ 2 p_ 2+b_ 3 p_ 3=n \] has prime solutions \(p_ 1,p_ 2,p_ 3\) satisfying max \(p_ j<B^ A\) where \(B=\max \{B_ 0, | b_ 1|, | b_ 2|, | b_ 3| \}.\) It is easy to see that the constant A cannot be replaced by 1. So, irrespective to the value of A, the above result gives the best possible bound for prime solutions of this type of equations. The original work on this problem can be traced back to a paper due to \textit{A. Baker} [J. Reine Angew. Math. 228, 166-181 (1967; Zbl 0155.092)].
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11N35
zbMATH DE Number: 3970859
zbMATH Keywords: ternary equations
zbMATH Keywords: small prime solutions for linear equations
zbMATH Keywords: exceptional set in generalized Goldbach problem
zbMATH Keywords: circle method
zbMATH Keywords: linear sieve
author: Ming-Chit Liu
MaRDI profile type: Publication
cites work: On some diophantine inequalities involving primes.
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cites work: A bound for prime solutions of some ternary equations
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description: scientific article; zbMATH DE number 3970859