On the Waring-Goldbach problem for seventh powers

Author:
Angel V. Kumchev

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2927-2937

MSC (2000):
Primary 11P32, 11L20, 11N36, 11P05, 11P55

Published electronically:
April 25, 2005

MathSciNet review:
2159771

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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, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR**1790423****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****3.**Glyn Harman,*On the distribution of 𝛼𝑝 modulo one. II*, Proc. London Math. Soc. (3)**72**(1996), no. 2, 241–260. MR**1367078**, 10.1112/plms/s3-72.2.241**4.**L. K. Hua,*Some results in prime number theory*, Quart. J. Math. Oxford Ser.**9**(1938), 68-80.**5.**L. K. Hua,*Additive theory of prime numbers*, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, R.I., 1965. MR**0194404****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****7.**Koichi Kawada and Trevor D. Wooley,*On the Waring-Goldbach problem for fourth and fifth powers*, Proc. London Math. Soc. (3)**83**(2001), no. 1, 1–50. MR**1829558**, 10.1112/S0024611500012636**8.**A. Kumchev,*On the Waring-Goldbach 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****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****13.**K. Thanigasalam,*On admissible exponents for 𝑘th powers*, Bull. Calcutta Math. Soc.**86**(1994), no. 2, 175–178. MR**1323498****14.**R. C. Vaughan,*On Waring’s problem for smaller exponents*, Proc. London Math. Soc. (3)**52**(1986), no. 3, 445–463. MR**833645**, 10.1112/plms/s3-52.3.445**15.**R. C. Vaughan,*The Hardy-Littlewood method*, 2nd ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997. MR**1435742****16.**I. M. Vinogradov,*Representation of an odd number as the sum of three primes*, Dokl. Akad. Nauk SSSR**15**(1937), 291-294, in Russian.

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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:
https://doi.org/10.1090/S0002-9939-05-07908-6

Received by editor(s):
May 17, 2004

Received by editor(s) in revised form:
June 10, 2004

Published electronically:
April 25, 2005

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.