On the WaringGoldbach problem for seventh powers
Author:
Angel V. Kumchev
Journal:
Proc. Amer. Math. Soc. 133 (2005), 29272937
MSC (2000):
Primary 11P32, 11L20, 11N36, 11P05, 11P55
Published electronically:
April 25, 2005
MathSciNet review:
2159771
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of seventh powers of prime numbers.
 1.
Harold
Davenport, Multiplicative number theory, 3rd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000.
Revised and with a preface by Hugh L. Montgomery. MR 1790423
(2001f:11001)
 2.
N.
G. De Bruijn, On the number of uncancelled elements in the sieve of
Eratosthenes, Nederl. Akad. Wetensch., Proc. 53
(1950), 803–812 = Indagationes Math. 12, 247–256 (1950). MR 0035785
(12,11d)
 3.
Glyn
Harman, On the distribution of 𝛼𝑝 modulo one.
II, Proc. London Math. Soc. (3) 72 (1996),
no. 2, 241–260. MR 1367078
(96k:11089), http://dx.doi.org/10.1112/plms/s372.2.241
 4.
L. K. Hua, Some results in prime number theory, Quart. J. Math. Oxford Ser. 9 (1938), 6880.
 5.
L.
K. Hua, Additive theory of prime numbers, Translations of
Mathematical Monographs, Vol. 13, American Mathematical Society,
Providence, R.I., 1965. MR 0194404
(33 #2614)
 6.
A.
E. Ingham, The distribution of prime numbers, Cambridge
Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint
of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
(91f:11064)
 7.
Koichi
Kawada and Trevor
D. Wooley, On the WaringGoldbach problem for fourth and fifth
powers, Proc. London Math. Soc. (3) 83 (2001),
no. 1, 1–50. MR 1829558
(2002b:11134), http://dx.doi.org/10.1112/S0024611500012636
 8.
A. Kumchev, On the WaringGoldbach problem. Exceptional sets for sums of cubes and higher powers, to appear in Canad. J. Math.
 9.
, On Weyl sums over primes and almost primes, preprint.
 10.
J. Y. Liu and T. Zhan, The exceptional set in Hua's theorem for three squares of primes, to appear in Acta. Math. Sinica.
 11.
K.
Thanigasalam, Improvement on Davenport’s iterative method and
new results in additive number theory. I, Acta Arith.
46 (1985), no. 1, 1–31. MR 831261
(87e:11118)
 12.
K.
Thanigasalam, Improvement on Davenport’s iterative method and
new results in additive number theory. III, Acta Arith.
48 (1987), no. 2, 97–116. MR 895435
(88f:11097)
 13.
K.
Thanigasalam, On admissible exponents for 𝑘th powers,
Bull. Calcutta Math. Soc. 86 (1994), no. 2,
175–178. MR 1323498
(96c:11117)
 14.
R.
C. Vaughan, On Waring’s problem for smaller exponents,
Proc. London Math. Soc. (3) 52 (1986), no. 3,
445–463. MR
833645 (87g:11126), http://dx.doi.org/10.1112/plms/s352.3.445
 15.
R.
C. Vaughan, The HardyLittlewood method, 2nd ed., Cambridge
Tracts in Mathematics, vol. 125, Cambridge University Press,
Cambridge, 1997. MR 1435742
(98a:11133)
 16.
I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 291294, in Russian.
 1.
 H. Davenport, Multiplicative Number Theory, third ed., Graduate Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000, revised by H. L. Montgomery. MR 1790423 (2001f:11001)
 2.
 N. J. de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Proc. Kon. Ned. Akad. Wetensch. 53 (1950), 803812. MR 0035785 (12:11d)
 3.
 G. Harman, On the distribution of modulo one II, Proc. London Math. Soc. (3) 72 (1996), 241260. MR 1367078 (96k:11089)
 4.
 L. K. Hua, Some results in prime number theory, Quart. J. Math. Oxford Ser. 9 (1938), 6880.
 5.
 , Additive Theory of Prime Numbers, American Mathematical Society, Providence, RI, 1965. MR 0194404 (33:2614)
 6.
 A. E. Ingham, The Distribution of Primes, reprint of the 1932 original ed., Cambridge University Press, Cambridge, 1990, with a foreword by R. C. Vaughan. MR 1074573 (91f:11064)
 7.
 K. Kawada and T. D. Wooley, On the WaringGoldbach problem for fourth and fifth powers, Proc. London Math. Soc. (3) 83 (2001), 150. MR 1829558 (2002b:11134)
 8.
 A. Kumchev, On the WaringGoldbach problem. Exceptional sets for sums of cubes and higher powers, to appear in Canad. J. Math.
 9.
 , On Weyl sums over primes and almost primes, preprint.
 10.
 J. Y. Liu and T. Zhan, The exceptional set in Hua's theorem for three squares of primes, to appear in Acta. Math. Sinica.
 11.
 K. Thanigasalam, Improvement on Davenport's iterative method and new results in additive number theory I, Acta Arith. 46 (1985), 131. MR 0831261 (87e:11118)
 12.
 , Improvement on Davenport's iterative method and new results in additive number theory III, Acta Arith. 48 (1987), 97116. MR 0895435 (88f:11097)
 13.
 , On admissible exponents for th powers, Bull. Calcutta Math. Soc. 86 (1994), 175178. MR 1323498 (96c:11117)
 14.
 R. C. Vaughan, On Waring's problem for smaller exponents, Proc. London Math. Soc. (3) 52 (1986), 445463. MR 0833645 (87g:11126)
 15.
 , The HardyLittlewood Method, second ed., Cambridge Tracts Math., vol. 125, Cambridge University Press, Cambridge, 1997. MR 1435742 (98a:11133)
 16.
 I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 291294, in Russian.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11P32,
11L20,
11N36,
11P05,
11P55
Retrieve articles in all journals
with MSC (2000):
11P32,
11L20,
11N36,
11P05,
11P55
Additional Information
Angel V. Kumchev
Affiliation:
Department of Mathematics, 1 University Station, C1200, The University of Texas at Austin, Austin, Texas 78712
Email:
kumchev@math.utexas.edu
DOI:
http://dx.doi.org/10.1090/S0002993905079086
PII:
S 00029939(05)079086
Received by editor(s):
May 17, 2004
Received by editor(s) in revised form:
June 10, 2004
Published electronically:
April 25, 2005
Communicated by:
WenChing Winnie Li
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
