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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
Citations:
Zbl 1238.11047
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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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