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Diophantine approximation with four squares and one $$k$$th power of primes. (English) Zbl 1396.11059
Summary: Let $$k$$ be an integer with $$k\geq 3$$ and $$\eta$$ be any real number. Suppose that $$\lambda_1,\lambda_2,\lambda_3,\lambda_4,\mu$$ are non-zero real numbers, not all of the same sign and $$\lambda_1/\lambda_2$$ is irrational. It is proved that the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\mu p_5^k+\eta|<(\max p_j)^{-\sigma}$ has infinitely many solutions in prime variables $$p_1,p_2,\dots,p_5$$, where $$0<\sigma<\frac{1}{16}$$ for $$k=3$$, $$0<\sigma<\frac{5}{3k2^k}$$ for $$4\leq k\leq 5$$ and $$0<\sigma<\frac{40}{21k2^k}$$ for $$k\geq 6$$. This gives an improvement of an earlier result [W. Li and W. Wang, J. Math. Sci. Adv. Appl. 6, No. 1, 1–16 (2010; Zbl 1238.11047)].

##### MSC:
 11D75 Diophantine inequalities 11P32 Goldbach-type theorems; other additive questions involving primes
##### Keywords:
Diophantine inequalities; Davenport-Heilbronn method; prime
Zbl 1238.11047
Full Text:
##### References:
 [1] Cook, RJ, The value of additive forms at prime arguments, Journal de Théorie des Nombres de Bordeaux, 13, 77-91, (2001) · Zbl 1047.11095 [2] Davenport, H; Heilbronn, H, On indefinite quadratic forms in five variables, J. Lond. Math. Soc., 21, 185-193, (1946) · Zbl 0060.11914 [3] Ghosh, A, The distribution of $$α p^2$$ modulo one, Proc. Lond. Math. Soc., 42, 252-269, (1981) · Zbl 0447.10035 [4] Harman, G, The values of ternary quadratic forms at prime arguments, Mathematika, 51, 83-96, (2004) · Zbl 1107.11043 [5] Hua, LK, Some results in additive prime number theory, Q. J. Math. (Oxford), 9, 68-80, (1938) · JFM 64.0131.02 [6] Languasco, A; Zaccagnini, A, A Diophantine problem with a prime and three squares of primes, J. Number Theory, 132, 3016-3028, (2012) · Zbl 1306.11032 [7] Languasco, A., Zaccagnini, A.: A diophantine problem with prime variables. arXiv:1206.0252 (2012) · Zbl 1306.11032 [8] Languasco, A; Settimi, V, On a Diophantine problem with one prime, two squares of primes and $$s$$ powers of two, Acta Arith., 154, 385-412, (2012) · Zbl 1306.11031 [9] Li, WP; Wang, TZ, Diophantine approximation with four squares and one $$k$$-th power of primes, J. Math. Sci. Adv. Appl., 6, 1-16, (2010) · Zbl 1238.11047 [10] Li, WP; Wang, TZ, Diophantine approximation with one prime and three squares of primes, Ramanujan J., 25, 343-357, (2011) · Zbl 1234.11036 [11] Liu, ZX; Sun, HW, Diophantine approximation with one prime and three squares of primes, Ramanujan J., 30, 327-340, (2013) · Zbl 1281.11029 [12] Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn (Revised by Heath-Brown, D.R.). Oxford University Press, Oxford (1986) · Zbl 0601.10026 [13] Vaughan, R.C.: The Hardy-Littlewood method, 2nd edn. Cambridge University Press, Cambridge (1997) · Zbl 0868.11046 [14] Watson, GL, On indefinite quadratic forms in five variables, Proc. Lond. Math. Soc., 3, 170-181, (1953) · Zbl 0050.04704
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