Two results on Goldbach-Linnik problems for cubes of primes. (English) Zbl 1495.11118

Liu, Yuhui

Rocky Mt. J. Math. 52, No. 3, 999-1007 (2022).

Author’s abstract: It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations of eight prime cubes and 609 powers of 2, and each sufficiently large even integer is the sum of eight cubes of primes and 157 powers of 2. These results constitute refinements upon those of Z. Liu [J. Number Theory 133, No. 10, 3339–3347 (2013; Zbl 1295.11109)] and of X. Zhao and W. Ge [Int. J. Number Theory 16, No. 7, 1547–1555 (2020; Zbl 1470.11257)].

Reviewer: Giovanni Coppola (Napoli)

MSC:

11P32	Goldbach-type theorems; other additive questions involving primes
11P55	Applications of the Hardy-Littlewood method

Keywords:

Waring-Goldbach problem; Hardy-Littlewood method; additive number theory

Citations:

Zbl 1295.11109; Zbl 1470.11257

References:

[1]	C. Elsholtz and J.-C. Schlage-Puchta, “The density of integers representable as the sum of four prime cubes”, Acta Arith. 192:4 (2020), 363-369. · Zbl 1445.11109 · doi:10.4064/aa180827-26-2
[2]	L.-K. Hua, “Some results in the additive prime-number theory”, Quart. J. Math. Oxford Ser. (2) 9:1 (1938), 68-80. · Zbl 0018.29404 · doi:10.1093/qmath/os-9.1.68
[3]	K. Kawada, “Note on the sum of cubes of primes and an almost prime”, Arch. Math. (Basel) 69:1 (1997), 13-19. · Zbl 0882.11057 · doi:10.1007/s000130050088
[4]	Z. Liu, “On pairs of quadratic equations in primes and powers of 2”, J. Number Theory 133:10 (2013), 3339-3347. · Zbl 1295.11109 · doi:10.1016/j.jnt.2013.04.006
[5]	J. Liu and M.-C. Liu, “Representation of even integers by cubes of primes and powers of 2”, Acta Math. Hungar. 91:3 (2001), 217-243. · Zbl 0980.11045
[6]	Z. Liu and G. Lü, “Eight cubes of primes and powers of 2”, Acta Arith. 145:2 (2010), 171-192. · Zbl 1239.11109 · doi:10.4064/aa145-2-6
[7]	D. J. Platt and T. S. Trudgian, “Linnik’s approximation to Goldbach’s conjecture, and other problems”, J. Number Theory 153 (2015), 54-62. · Zbl 1328.11102 · doi:10.1016/j.jnt.2015.01.008
[8]	X. Ren, “Sums of four cubes of primes”, J. Number Theory 98:1 (2003), 156-171. · Zbl 1100.11033 · doi:10.1016/S0022-314X(02)00022-7
[9]	K. F. Roth, “On Waring’s problem for cubes”, Proc. London Math. Soc. (2) 53 (1951), 268-279. · Zbl 0043.27303 · doi:10.1112/plms/s2-53.4.268
[10]	X. Zhao, “Goldbach-Linnik type problems on cubes of primes”, Ramanujan J. 57:1 (2022), 239-251. · Zbl 1498.11203 · doi:10.1007/s11139-020-00303-9
[11]	X. Zhao and W. Ge, “Eight cubes of primes and 204 powers of 2”, Int. J. Number Theory 16:7 (2020), 1547-1555. · Zbl 1470.11257 · doi:10.1142/S1793042120500803