On the Waring–Goldbach problem for seventh powers
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- by Angel V. Kumchev PDF
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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 $46$ seventh powers of prime numbers.References
- 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
- 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 35785
- Glyn Harman, On the distribution of $\alpha p$ modulo one. II, Proc. London Math. Soc. (3) 72 (1996), no. 2, 241–260. MR 1367078, DOI 10.1112/plms/s3-72.2.241
- L. K. Hua, Some results in prime number theory, Quart. J. Math. Oxford Ser. 9 (1938), 68–80.
- L. K. Hua, Additive theory of prime numbers, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, R.I., 1965. MR 0194404
- 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
- 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, DOI 10.1112/S0024611500012636
- A. Kumchev, On the Waring–Goldbach problem. Exceptional sets for sums of cubes and higher powers, to appear in Canad. J. Math.
- —, On Weyl sums over primes and almost primes, preprint.
- J. Y. Liu and T. Zhan, The exceptional set in Hua’s theorem for three squares of primes, to appear in Acta. Math. Sinica.
- 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, DOI 10.4064/aa-46-1-1-31
- 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, DOI 10.4064/aa-48-2-97-116
- K. Thanigasalam, On admissible exponents for $k$th powers, Bull. Calcutta Math. Soc. 86 (1994), no. 2, 175–178. MR 1323498
- R. C. Vaughan, On Waring’s problem for smaller exponents, Proc. London Math. Soc. (3) 52 (1986), no. 3, 445–463. MR 833645, DOI 10.1112/plms/s3-52.3.445
- R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997. MR 1435742, DOI 10.1017/CBO9780511470929
- I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 291–294, in Russian.
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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2927-2937
- MSC (2000): Primary 11P32, 11L20, 11N36, 11P05, 11P55
- DOI: https://doi.org/10.1090/S0002-9939-05-07908-6
- MathSciNet review: 2159771