Riv. Mat. Univ. Parma, Vol. 12, No. 1, 2021
Jörg Brüdern [a]
Roth's ascending powers in the primes
Pages: 29-40
Received: 2 February 2020
Accepted: 3 April 2020
Mathematics Subject Classification (2010): 11P55.
Keywords: Waring-Goldbach problems, Hardy-Littlewood method.
Author address:
[a]: Universität Göttingen, Mathematisches Institut, Bunsenstrasse 3-5,
D-37073 Göttingen, Germany
This research was partially supported by Deutsche Forschungsgemeinschaft.
Full Text (PDF)
Abstract:
The number of variables in Roth's ascending powers problem with prime variables is reduced by three.
References
- [1]
-
J. Brüdern, Sums of squares and higher powers, II,
J. London Math. Soc. (2) 35 (1987), 244-250.
MR1575093
- [2]
-
J. Brüdern,
A problem in additive number theory,
Math. Proc. Cambridge Philos. Soc. 103 (1988), 27-33.
MR0913447
- [3]
-
H. Davenport,
On sums of positive integral k-th powers,
Amer. J. Math. 64 (1942), 189-198.
MR0005498
- [4]
-
H. Davenport and P. Erdős,
On sums of positive integral \(k^{th}\) powers,
Ann. of Math. 40 (1939), 533-536.
MR0000027
- [5]
-
K. B. Ford,
The representation of numbers as sums of unlike powers,
J. London Math. Soc. (2) 51 (1995), 14-26.
MR1310718
- [6]
-
K. B. Ford,
The representation of numbers as sums of unlike powers, II,
J. Amer. Math. Soc. 9 (1996), 919-940.
MR1325794
- [7]
-
G. H. Hardy and J. E. Littlewood,
Some problems of 'Partitio Numerorum' (VI): Further researches in Waring's Problem,
Math. Z. 23 (1925), 1-37.
MR1544728
- [8]
-
G. Harman, Trigonometric sums over primes, I,
Mathematika 28 (1981), 249-254.
MR0645105
- [9]
-
L.-K. Hua,
Some results in the additive prime-number theory,
Quart. J. Math. Oxford Ser. (2) 9 (1938), 68-80.
MR3363459
- [10]
-
A. V. Kumchev,
On Weyl sums over primes and almost primes,
Michigan Math. J. 54 (2006), 243-268.
MR2252758
- [11]
-
K. F. Roth,
Proof that almost all positive integers are sums of a square, a positive cube and a fourth power,
J. London Math. Soc. 24 (1949), 4-13.
MR0028336
- [12]
-
K. F. Roth,
A problem in additive number theory,
Proc. London Math. Soc. (2) 53 (1951), 381-395.
MR0041874
- [13]
-
K. Thanigasalam,
On additive number theory,
Acta Arith. 13 (1967/1968), 237-258.
MR0222044
- [14]
-
K. Thanigasalam,
On sums of powers and a related problem,
Acta Arith. 36 (1980), 125-141.
MR0581911
- [15]
-
K. Thanigasalam,
On certain additive representations of integers,
Portugal. Math. 42 (1983/84), 447-465.
MR0836123
- [16]
-
R. C. Vaughan,
On the representation of numbers as sums of powers of natural numbers,
Proc. London Math. Soc. (3) 21 (1970), 160-80.
MR0272734
- [17]
-
R. C. Vaughan,
On sums of mixed powers,
J. London Math. Soc. (2) 3 (1971), 677-688.
MR0294279
- [18]
-
R. C. Vaughan,
On Waring's problem for smaller exponents,
Proc. London Math. Soc. (3) 52 (1986), 445-463.
MR0833645
- [19]
-
R.C. Vaughan
The Hardy-Littlewood method, 2nd ed.,
University Press, Cambridge, 1997.
MR1435742
- [20]
-
L. Zhao,
The additive problem with one cube and three cubes of primes,
Michigan Math. J. 63 (2014), 763-779.
MR3286670
Home Riv.Mat.Univ.Parma