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Linear Equations with Small Prime and Almost Prime Solutions

Published online by Cambridge University Press:  20 November 2018

Xianmeng Meng
Xianmeng Meng*
Affiliation:
Department of Statistics and Mathematics, Shandong Finance Institute, Jinan, Shandong, 250014, P.R. China. e-mail: [email protected]
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Abstract

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Let ${{b}_{1}}$, ${{b}_{2}}$ be any integers such that $\gcd \left( {{b}_{1}},{{b}_{2}} \right)=1$ and ${{c}_{1}}\left| {{b}_{1}} \right|\,<\,\left| {{b}_{2}} \right|\,\le \,{{c}_{2}}\left| {{b}_{1}} \right|$ , where ${{c}_{1}}$, ${{c}_{2}}$ are any given positive constants. Let $n$ be any integer satisfying $\gcd \left( n,\,{{b}_{i}} \right)\,=\,1$ , $i\,=\,1,\,2$. Let ${{P}_{k}}$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all ${{b}_{2}}$ , we prove (i) a sharp lower bound for $n$ such that the equation ${{b}_{1}}p\,+\,{{b}_{2}}m\,=\,n$ is solvable in prime $p$ and almost prime $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$ whenever both ${{b}_{i}}$ are positive, and (ii) a sharp upper bound for the least solutions $p$, $m$ of the above equation whenever ${{b}_{i}}$ are not of the same sign, where $p$ is a prime and $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$.

Keywords

11P3211N36sieve methodadditive problem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 3 , 01 September 2008 , pp. 399 - 405
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Baker, A., On some diophantine inequalities involving primes. J. Reine Angew. Math. 228(1967), 166–181.Google Scholar
[2] Chen, J.-R., On the representation of a larger even intger as the sum of a prime and the product of at most two primes. Sci. Sinica 16(1973), 157–176.Google Scholar
[3] Choi, K. K., Liu, M. C., and Tsang, K. M., Conditional bounds for small prime solutions of linear equations. Manuscripta Math. 74(1992), no. 3, 321–340.Google Scholar
[4] Coleman, M. D., On the equation b 1p – b 2 P 2 = b 3 . J. Reine Angew. Math. 403(1990), 1–66.Google Scholar
[5] Halberstam, H. and Richert, H. E., Sieve Methods. London Mathematical Society Monographs 4, London, Academic Press, 1974.Google Scholar
[6] Li, H. Z., Small prime solutions of linear ternary equations. Acta Arith. 98(2001), no. 3, 293–309.Google Scholar
[7] Liu, M. C., On binary equations. Monatsh. Math. 96(1983), no. 4, 271–276.Google Scholar
[8] Liu, M. C. and Tsang, K. M., Small prime solutions of linear equations. In: Théorie des Nombres. Walter de Gruyter, Berlin, 1989, pp. 595–624.Google Scholar
[9] Liu, M. C. and Wang, T. Z., A numerical bound for small prime solutions of some ternary linear equations. Acta Arith. 86(1998), no. 4, 343–383.Google Scholar
[10] Pan, C. D. and Pan, C. B., Goldbach Conjecture. Science Press, Beijing, 1992.Google Scholar