Diophantine Inequality by Unlike Powers of Primes

Abstract

Suppose that λ₁, ⋯ λ₅ are nonzero real numbers, not all of the same sign, satisfying that ${{{\lambda _1}} \over {{\lambda _2}}}$ is irrational. Then for any given real number η and ε > 0, the inequality$\left| {{\lambda _1}{p_1} + {\lambda _2}p_2^2 + {\lambda _3}p_3^3 + {\lambda _4}p_4^4 + {\lambda _5}p_5^5 + \eta } \right| < {\left( {\mathop {\max }\limits_{1 \le j \le 5} p_j^j} \right)^{ - {{19} \over {756}} + \varepsilon }}$

has infinitely many solutions in prime variables p ₁, ⋯, p ₅. This result constitutes an improvement of the recent results.

Keywords

Prime / Davenport-Heilbronn method / Diophantine inequalities

References

[1] Brüdern J. The Davenport-Heilbronn Fourier transform method and some Diophantine inequalities, Number Theory and its Applications (Kyoto, 1997), 1999, Dordrecht: Kluwer Acad. Publ. 59-87 Dev. Math., 2

[2] Ge, W. X. and Li, W. P., One diophantine inequality with unlike powers of prime variables, J. Inequal. Appl., 33, 2016, 8 pages.

[3] Ge W X, Zhao F. The values of cubic forms at prime arguments. J. Number Theory, 2017, 180: 694-709

[4] Harman G. The values of ternary quadratic forms at prime arguments. Mathematika, 2004, 51: 83-96

[5] Harman G, Kumchev A V. On sums of squares of primes. Math. Proc. Cambridge Philos. Soc., 2006, 140(1): 1-13

[6] Kumchev A V. On Weyl sums over primes and almost primes. Michigan Math. J., 2006, 54: 243-268

[7] Languasco A, Zaccagnini A. A Diophantine problem with a prime and three squares of primes. J. Number Theory, 2012, 132: 3016-3028

[8] Liu Z X. Diophantine approximation by unlike powers of primes. Int. J. Number Theory, 2017, 13: 2445-2452

[9] Mu Q W. One diophantine inequality with unlike powers of prime variables. Int. J. Number Theory, 2017, 13: 1531-1545

[10] Mu Q W, Qu Y Y. A note on Diophantine approximation by unlike powers of primes. Int. J. Number Theory, 2018, 14: 1651-1668

[11] Prachar K. Über ein Problem vom Waring-Goldbach’schen Typ II. Monatsh. Math., 1953, 57: 113-116 in German)

[12] Wang Y C, Yao W L. Diophantine approximation with one prime and three squares of primes. J. Number Theory, 2017, 180: 234-250