Roth’s ascending powers in the primes. (English) Zbl 1480.11124

Brüdern, Jörg

Riv. Mat. Univ. Parma (N.S.) 12, No. 1, 29-40 (2021).

K. F. Roth [Proc. London Math. Soc. (2) 53, 381–395 (1951; Zbl 0044.03601)] proved that, for sufficiently large natural number \(n,\) the Diophantine equation \[n=\sum_{k=1}^{50} x_k^{k+1}\] has solutions in natural numbers \(x_j.\) The upper bound \(50\) was improved by many authors. The current record is \(14\) due to K. B. Ford [J. Amer. Math. Soc 9, 919–940 (1996; Zbl 0866.11054)].
R. C. Vaughan [J. London Math. Soc. (2) 3, 677–688 (1971; Zbl 0221.10050)] studied the equation \[n=\sum_{k=1}^s x_k^{k+1}\] with the variables \(x_j\) constrained to sequences that are not too thin. In particular, it follows from his work that when \(s\ge 30\) and \(n\equiv s \mod 2\) is large, then the equation \[n=\sum_{k=1}^s x_k^{k+1}\] has solutions in primes \(x_j.\) K. Thanigasalam [Acta Arith. 36, 125–141 (1980; Zbl 0354.10016)] obtained the same result when \(s\ge 23.\)
In the paper under review the author proves the above result when \(s=20.\) The proof employs the Hardy-Littlewood method and a recent work of L. Zhao [Michigan Math. J. 63, 763–779 (2014; Zbl 1360.11092)].

Reviewer: Mihály Szalay (Budapest)

MSC:

11P55	Applications of the Hardy-Littlewood method
11D72	Diophantine equations in many variables
11D85	Representation problems
11P05	Waring’s problem and variants
11P32	Goldbach-type theorems; other additive questions involving primes

Keywords:

Waring-Goldbach problems; Hardy-Littlewood method

Citations:

Zbl 0044.03601; Zbl 0866.11054; Zbl 0221.10050; Zbl 0354.10016; Zbl 1360.11092