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Small Prime Solutions to Cubic Diophantine Equations

Published online by Cambridge University Press:  20 November 2018

Zhixin Liu
Zhixin Liu*
Affiliation:
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P. R. China e-mail: zhixinliu@tju.edu.cn
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Abstract.

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Let ${{a}_{1}},...,{{a}_{9}}$ be nonzero integers and $n$ any integer. Suppose that ${{a}_{1}}+\cdot \cdot \cdot +{{a}_{9}}\,\equiv \,n\,\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{j}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\,\le \,9$. In this paper we prove the following:

(i) If ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$.

(ii) If all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$, then ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ is solvable in primes ${{p}_{j}}$.

These results are an extension of the linear and quadratic relative problems.

Keywords

11P3211P0511P55small primeWaring–Goldbach problemcircle method
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 785 - 794
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Baker, A., On some diophantine inequalities involving primes. J. Reine Angew. Math. 228 (1967), 166–181.Google Scholar
[2] Choi, K. K., A numerical bound for Baker's constant—some explicit estimates for small prime solutions of linear equations. Bull. Hong Kong Math. Soc. 1 (1997), 1–19.Google Scholar
[3] Choi, K. K. and Kumchev, A. V., Mean values of Dirichlet polynomials and applications to linear equations with prime variables. Acta Arith. 123 (2006), 125–142. http://dx.doi.org/10.4064/aa123-2-2 Google Scholar
[4] Choi, K. K., Quadratic equations with five prime unknowns. J. Number Theor. 107 (2004), 357–367. http://dx.doi.org/10.1016/j.jnt.2004.03.007 Google Scholar
[5] Choi, K. K. and Liu, J. Y., Small prime solutions of quadratic equations II. Proc. Amer. Math. Soc. 133 (2005), 945–951. http://dx.doi.org/10.1090/S0002-9939-04-07784-6 Google Scholar
[6] Choi, K. K., Small prime solutions of quadratic equations. Canad. J. Math. 54 (2002), 71–91. http://dx.doi.org/10.4153/CJM-2002-004-4 Google Scholar
[7] Choi, K. K., Liu, M. C., and Tsang, K. M., Conditional bounds for small prime solutions of linear equations. Manuscripta Math. 74 (1992), 321–340. http://dx.doi.org/10.1007/BF02567674 Google Scholar
[8] Harman, G. and Kumchev, A. V., On sums of squares of primes. Math. Proc. Cambridge Philos. Soc. 140 (2006), 1–13. http://dx.doi.org/10.1017/S0305004105008819 Google Scholar
[9] Hua, L. K., Some results in the additive prime number theory. Quart. J. Math. (Oxford) 9 (1938), 68–80.Google Scholar
[10] Kumchev, A. V., On Weyl sums over primes and almost primes. Michigan Math. J. 54 (2006), 243–268. http://dx.doi.org/10.1307/mmj/1156345592 Google Scholar
[11] Leung, D., Small prime solutions to cubic Diophantine equations. Master's thesis, Simon Fraser University, 2006.Google Scholar
[12] Li, H. Z., Small prime solutions of some ternary linear equations. Acta Arith. 98 (2001), 293–309. http://dx.doi.org/10.4064/aa98-3-6 Google Scholar
[13] Liu, M. C. and Tsang, K. M., Small prime solutions of some additive equations. Monatsh. Math. 111 (1991), 147–169. http://dx.doi.org/10.1007/BF01332353 Google Scholar
[14] Liu, M. C., Small prime solutions of linear equations. In: Théorie des nombres (Québec, PQ, 1987), de Gruyter, Berlin, 1989, 595–624.Google Scholar
[15] Liu, M. C. and T. Z.Wang, A numerical bound for small prime solutions of some ternary linear equations. Acta Arith. 86 (1998), 343–383.Google Scholar
[16] Liu, J. Y., Enlarged major arcs in additive problems, II. Proc. Steklov Inst. Math. 276 (2012), 176–192.Google Scholar
[17] Liu, J. Y., On Lagrange's theorem with prime variables. Quart. J. Math. (Oxford) 54 (2003), 453–462. http://dx.doi.org/10.1093/qmath/hag028 Google Scholar
[18] Liu, J. Y. and Tsang, K. M., Small prime solutions of ternary linear equations. Acta Arith. 118 (2005), 79–100. http://dx.doi.org/10.4064/aa118-1-5 Google Scholar
[19] Ren, X. M., On exponential sums over primes and application in Waring–Goldbach problem. Sci. China Ser. A 48 (2005), 785–797. http://dx.doi.org/10.1360/03ys0341 Google Scholar