On the Equation x21 + x22 + x23 + x24 = N with Variables such that x1x2x3x4 + 1 is an Almost-Prime
By: T. L. Todorova and D. I. Tolev
References
- [1] BLOMER, V.-BRÜDERN, J.: A three squares theorem with almost primes, Bull. London Math. Soc. 37 (2005), 507-513.
- [2] BRÜDERN, J.-FOUVRY, E.: Lagranges four squares theorem with almost prime variables, J. Reine Angew. Math. 454 (1994), 59-96.
- [3] ESTERMANN, T.: A new application of the Hardy-Littlewood-Kloosterman method, Proc. London Math. Soc. 12 (1962), 425-444.
- [4] FRIEDLANDER, J.-IWANIEC, H.: Opera de Cribro, in: Amer. Math. Soc. Colloq. Publ., Vol. 57, Providence, RI, 2010.
- [5] GRIEVES, G.: On the representation of a number in the form x+ y+ p+ q, where p and q are odd primes, Acta Arith. 29 (1976), 257-274.
- [6] HARDY, G. H.-WRIGHT, E. M.: An Introduction to the Theory of Numbers (5ed.), Oxford Univ. Press, 1979.
- [7] HEATH-BROWN, D. R.: Cubic forms in ten variables, Proc. London Math. Soc. 47 (1983), 225-257.
- [8] HEATH-BROWN, D. R.-TOLEV, D. I.: Lagranges four squares theorem with one prime and three almost-prime variables, J. Reine Angew. Math. 558 (2003), 159-224.
- [9] HUA, L. K.: Introduction to Number Theory. Springer, Berlin, 1982.
- [10] IWANIEC, H.-KOWALSKI, E.: Analytic Number Theory, in: Amer. Math. Soc. Colloq. Publ., Vol. 53, Providence, RI, 2004.
- [11] KARATSUBA, A. A.: Basic Analytic Number Theory. Springer, Berlin, 1993.
- [12] KLOOSTERMAN, H. D.: On the representation of numbers in the form ax+ by+ cz+ dt, Acta Math. 49 (1926), 407-464.
- [13] KOWALCHIK, F. B.: Analogues of the Hardy-Litlewood equation, Zap. Nauchn. Sem. LOMI 116 (1982), 86-95.
- [14] LÜ, G.: Gauss’s three squares theorem with almost prime variables, Acta Arith. 128 (2007), 391-399.
- [15] PLAKSIN, V. A.: An asymptotic formula for the number of solutions of a nonlinear equation for prime numbers, Math. USSR Izv. 18 (1982), 275-348.
- [16] SHIELDS, P.: Some Applications of the Sieve Methods in Number Theory. Thesis, University of Wales, Cardiff, UK, 1979.
- [17] TOLEV, D. I.: Lagranges four squares theorem with variables of special type, in: Proceedings of the Session in Analytic Number Theory and Diophantine Equations (D. R. Heath- -Brown et al., eds.), Bonn, 2002, Bonner Math. Schriften, Vol. 360, Univ. Bonn, Bonn, 2003, 17 pp.
- [18] CAI, Y.: Lagranges four squares theorem with variables of special type, Intern. J. Number Theory 6 (2010), 1801-1817.
DOI: https://doi.org/10.2478/tmmp-2014-0015 | Journal eISSN: 1338-9750 (formerly 1210-3195) | Journal ISSN: 1210-3195
Language: English
Page range: 1 - 26
Submitted on: Jun 22, 2014
Published on: Mar 11, 2015
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
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© 2015 T. L. Todorova, D. I. Tolev, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.