DOI QR코드

DOI QR Code

DIOPHANTINE INEQUALITY WITH FOUR SQUARES AND ONE kTH POWER OF PRIMES

Zhu, Li

  • Received : 2018.07.22
  • Accepted : 2019.03.04
  • Published : 2019.07.01

Abstract

Let k be an integer with $k{\geq}3$. Define $h(k)=[{\frac{k+1}{2}}]$, ${\sigma}(k)={\min}\(2^{h(k)-1},\;{\frac{1}{2}}h(k)(h(k)+1)\)$. Suppose that ${\lambda}_1,{\ldots},{\lambda}_5$ are non-zero real numbers, not all of the same sign, satisfying that ${\frac{{\lambda}_1}{{\lambda}_2}}$ is irrational. Then for any given real number ${\eta}$ and ${\varepsilon}>0$, the inequality $${\mid}{\lambda}_1p^2_1+{\lambda}_2p^2_2+{\lambda}_3p^2_3+{\lambda}_4p^2_4+{\lambda}_5p^k_5+{\eta}{\mid}<({\max_{1{\leq}j{\leq}5}}p_j)^{-{\frac{3}{20{\sigma}(k)}}+{\varepsilon}}$$ has infinitely many solutions in prime variables $p_1,{\ldots},p_5$. This gives an improvement of the recent results.

Keywords

prime;Davenport-Heilbronn method;sieve theory

References

  1. J. Brudern, The Davenport-Heilbronn Fourier transform method, and some Diophantine inequalities, in Number theory and its applications (Kyoto, 1997), 59-87, Dev. Math., 2, Kluwer Acad. Publ., Dordrecht, 1999.
  2. W. Ge and T. Wang, On Diophantine problems with mixed powers of primes, Acta Arith. 182 (2018), no. 2, 183-199.
  3. G. Harman, The values of ternary quadratic forms at prime arguments, Mathematika 51 (2004), no. 1-2, 83-96 (2005).
  4. G. Harman and A. Kumchev, On sums of squares of primes II, J. Number Theory 130 (2010), no. 9, 1969-2002. https://doi.org/10.1016/j.jnt.2010.03.010
  5. L.-K. Hua, Some results in the additive prime-number theory, Quart. J. Math. Oxford Ser. (2) 9 (1938), no. 1, 68-80.
  6. A. Languasco and A. Zaccagnini, A Diophantine problem with a prime and three squares of primes, J. Number Theory 132 (2012), no. 12, 3016-3028. https://doi.org/10.1016/j.jnt.2012.06.015
  7. W. Li and T. Wang, Diophantine approximation with four squares and one Kth power of primes, J. Math. Sci. Adv. Appl. 6 (2010), no. 1, 1-16.
  8. Z. Liu, Diophantine approximation by unlike powers of primes, Int. J. Number Theory 13 (2017), no. 9, 2445-2452. https://doi.org/10.1142/S1793042117501330
  9. Q. Mu, Diophantine approximation with four squares and one kth power of primes, Ramanujan J. 39 (2016), no. 3, 481-496.
  10. Q. Mu and Y. Qu, A note on Diophantine approximation by unlike powers of primes, Int. J. Number Theory 14 (2018), no. 6, 1651-1668.
  11. R. C. Vaughan, The Hardy-Littlewood Method, second edition, Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997.
  12. Y. Wang and W. Yao, Diophantine approximation with one prime and three squares of primes, J. Number Theory 180 (2017), 234-250.

Acknowledgement

Supported by : National Natural Science Foundation of China