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Keywords:
Diophantine equation; exponential sum; prime
Diophantine equation; exponential sum; prime
Summary:
Let $[\theta ]$ denote the integral part of the real number $\theta .$ We prove that for $1<c<\frac {25 908}{18 905}$, the Diophantine equation $$ [p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]+[p_{4}^{c}]+[p_{5}^{c}]=N $$ is solvable in prime variables $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ such that $p_1=x^2+y^2+1$ with integers $x$ and $y$ for sufficiently large integer $N$, and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023).
References:
[1] Baker, R.: Some Diophantine equations and inequalities with primes. Funct. Approximatio, Comment. Math. 64 (2021), 203-250. DOI 10.7169/facm/1912 | MR 4278752 | Zbl 1484.11195
[2] Buriev, K.: Additive Problems with Prime Numbers: Thesis. Moscow State University, Moscow (1989), Russian. MR 0946769
[3] Dimitrov, S.: A ternary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$. Ramanujan J. 59 (2022), 571-607. DOI 10.1007/s11139-021-00545-1 | MR 4480301 | Zbl 1497.11088
[4] Dimitrov, S.: On an equation by primes with one Linnik prime. Georgian Math. J. 29 (2022), 455-470. DOI 10.1515/gmj-2022-2144 | MR 4431316 | Zbl 1487.11079
[5] Dimitrov, S.: A Diophantine equation involving special prime numbers. Czech. Math. J. 73 (2023), 151-176. DOI 10.21136/CMJ.2022.0469-21 | MR 4541094 | Zbl 1538.11172
[6] Dimitrov, S.: A quinary Diophantine inequality by primes with one of the form $p=x^2+y^2+1$. Indian J. Pure Appl. Math. 55 (2024), 168-188. DOI 10.1007/s13226-022-00354-2 | MR 4703140 | Zbl 1535.11137
[7] Graham, S. W., Kolesnik, G.: Van Der Corput's Method of Exponential Sums. London Mathematical Society Lecture Note Series 126. Cambridge University Press, Cambridge (1991). DOI 10.1017/CBO9780511661976 | MR 1145488 | Zbl 0713.11001
[8] Heath-Brown, D. R.: The Pjateckii-Šapiro prime number theorem. J. Number Theory 16 (1983), 242-266. DOI 10.1016/0022-314X(83)90044-6 | MR 0698168 | Zbl 0513.10042
[9] Hua, L.-K.: Some results in the additive prime-number theory. Q. J. Math., Oxf. Ser. 9 (1938), 68-80. DOI 10.1093/qmath/os-9.1.68 | MR 3363459 | Zbl 0018.29404
[10] Huxley, M. N.: Exponential sums and the Riemann zeta function. V. Proc. Lond. Math. Soc., III. Ser. 90 (2005), 1-41. DOI 10.1112/S0024611504014959 | MR 2107036 | Zbl 1083.11052
[11] Li, S.: On a Diophantine equation with prime numbers. Int. J. Number Theory 15 (2019), 1601-1616. DOI 10.1142/S1793042119300011 | MR 3994149 | Zbl 1462.11084
[12] Linnik, Y. V.: An asymptotic formula in an additive problem of Hardy-Littlewood. Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 629-706 Russian. MR 0122796 | Zbl 0099.03501
[13] Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591 (2006), 1-20. DOI 10.1515/CRELLE.2006.012 | MR 2212877 | Zbl 1165.11067
[14] Sargos, P., Wu, J.: Multiple exponential sums with monomials and their applications in number theory. Acta Math. Hung. 87 (2000), 333-354. DOI 10.1023/A:1006777803163 | MR 1771211 | Zbl 0963.11045
[15] Zhang, M., Li, J.: On a Diophantine equation with five prime variables. Available at https://arxiv.org/abs/1809.04591v2 (2019), 17 pages. DOI 10.48550/arXiv.1809.04591
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