Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 73, No. 4, pp. 1219-1228, 2023

On pairs of Goldbach-Linnik equations with unequal powers of primes

Enxun Huang

Received April 21, 2023. Published online October 12, 2023.

Abstract: It is proved that every pair of sufficiently large odd integers can be represented by a pair of equations, each containing two squares of primes, two cubes of primes, two fourth powers of primes and 105 powers of 2.

Keywords: Goldbach-Waring-Linnik problem; circle method; powers of 2

Classification MSC: 11P32, 11P05, 11P55

DOI: 10.21136/CMJ.2023.0470-22

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] D. R. Heath-Brown, J.-C. Puchta: Integers represented as a sum of primes and powers of two. Asian J. Math. 6 (2002), 535-565. DOI 10.4310/AJM.2002.v6.n3.a7 | MR 1946346 | Zbl 1097.11050
[2] Y. Kong: On pairs of linear equations in four prime variables and powers of two. Bull. Aust. Math. Soc. 87 (2013), 55-67. DOI 10.1017/S0004972712000172 | MR 3011941 | Zbl 1266.11110
[3] Y. Kong, Z. Liu: On pairs of Goldbach-Linnik equations. Bull. Aust. Math. Soc. 95 (2017), 199-208. DOI 10.1017/S000497271600071X | MR 3614942 | Zbl 1381.11098
[4] A. V. Kumchev: On Weyl sums over primes and almost primes. Mich. Math. J. 54 (2006), 243-268. DOI 10.1307/mmj/1156345592 | MR 2252758 | Zbl 1137.11054
[5] A. Languasco, A. Zaccagnini: On a Diophantine problem with two primes and $s$ powers of two. Acta Arith. 145 (2010), 193-208. DOI 10.4064/aa145-2-7 | MR 2733083 | Zbl 1222.11049
[6] Y. V. Linnik: Prime numbers and powers of two. Tr. Mat. Inst. Steklova 38 (1951), 152-169. (In Russian.) MR 0050618 | Zbl 0049.31402
[7] Y. V. Linnik: Addition of prime numbers with powers of one and the same number. Mat. Sb., N. Ser. 32 (1953), 3-60. (In Russian.) MR 0059938 | Zbl 0051.03402
[8] J. Liu: Enlarged major arcs in additive problems. II. Proc. Steklov Inst. Math. 276 (2012), 176-192. DOI 10.1134/S0081543812010154 | MR 2986119 | Zbl 1297.11130
[9] Z. Liu: Goldbach-Linnik type problems with unequal powers of primes. J. Number Theory 176 (2017), 439-448. DOI 10.1016/j.jnt.2016.12.009 | MR 3622138 | Zbl 1422.11207
[10] J. Liu, M.-C. Liu, T. Wang: On the almost Goldbach problem of Linnik. J. Théor. Nombres Bordx. 11 (1999), 133-147. DOI 10.5802/jtnb.242 | MR 1730436 | Zbl 0979.11051
[11] X. Lü: On unequal powers of primes and powers of 2. Ramanujan J. 50 (2019), 111-121. DOI 10.1007/s11139-018-0128-2 | MR 4008100 | Zbl 1472.11269
[12] J. Pintz, I. Z. Ruzsa: On Linnik's approximation to Goldbach's problem. I. Acta. Arith. 109 (2003), 169-194. DOI 10.4064/aa109-2-6 | MR 1980645 | Zbl 1031.11060
[13] L. Zhao: On the Waring-Goldbach problem for fourth and sixth powers. Proc. Lond. Math. Soc. (3) 108 (2014), 1593-1622. DOI 10.1112/plms/pdt072 | MR 3218320 | Zbl 1370.11116
[14] X. Zhao: Goldbach-Linnik type problems on cubes of primes. Ramanujan J. 57 (2022), 239-251. DOI 10.1007/s11139-020-00303-9 | MR 4360484 | Zbl 1498.11203

Affiliations: Enxun Huang, School of Mathematics of Shandong University, 27 Shanda Nanlu, Jinan 250100, Shandong, P. R. China, e-mail: huangenxun@mail.sdu.edu.cn

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