Nous considérons une version affaiblie de la conjecture sur la représentation des entiers comme somme de quatre carrés de nombres premiers.
We consider an approximation to the popular conjecture about representations of integers as sums of four squares of prime numbers.
@article{JTNB_2005__17_3_925_0, author = {Tolev, Doychin}, title = {On the exceptional set of {Lagrange{\textquoteright}s} equation with three prime and one almost{\textendash}prime variables}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {925--948}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {3}, year = {2005}, doi = {10.5802/jtnb.528}, zbl = {05016595}, mrnumber = {2212133}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.528/} }
TY - JOUR AU - Tolev, Doychin TI - On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables JO - Journal de théorie des nombres de Bordeaux PY - 2005 SP - 925 EP - 948 VL - 17 IS - 3 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.528/ DO - 10.5802/jtnb.528 LA - en ID - JTNB_2005__17_3_925_0 ER -
%0 Journal Article %A Tolev, Doychin %T On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables %J Journal de théorie des nombres de Bordeaux %D 2005 %P 925-948 %V 17 %N 3 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.528/ %R 10.5802/jtnb.528 %G en %F JTNB_2005__17_3_925_0
Tolev, Doychin. On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 925-948. doi : 10.5802/jtnb.528. http://www.numdam.org/articles/10.5802/jtnb.528/
[1] C. Bauer, M.-C. Liu, T. Zhan, On a sum of three prime squares. J. Number Theory 85 (2000), 336–359. | MR | Zbl
[2] J. Brüdern, E. Fouvry, Lagrange’s Four Squares Theorem with almost prime variables. J. Reine Angew. Math. 454 (1994), 59–96. | MR | Zbl
[3] G. Greaves, On the representation of a number in the form where and are odd primes. Acta Arith. 29 (1976), 257–274. | MR | Zbl
[4] H. Halberstam, H.-E. Richert, Sieve methods. Academic Press, 1974. | MR | Zbl
[5] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers. Fifth ed., Oxford Univ. Press, 1979. | MR | Zbl
[6] G. Harman, A. V. Kumchev, On sums of squares of primes. Math. Proc. Cambridge Philos. Soc., to appear. | MR | Zbl
[7] D.R. Heath-Brown, Cubic forms in ten variables. Proc. London Math. Soc. 47 (1983), 225–257. | MR | Zbl
[8] D.R. Heath-Brown, D.I.Tolev, Lagrange’s four squares theorem with one prime and three almost–prime variables. J. Reine Angew. Math. 558 (2003), 159–224. | MR | Zbl
[9] L.K. Hua, Some results in the additive prime number theory. Quart. J. Math. Oxford 9 (1938), 68–80. | Zbl
[10] L.K. Hua, Introduction to number theory. Springer, 1982. | MR | Zbl
[11] L.K. Hua, Additive theory of prime numbers. American Mathematical Society, Providence, 1965. | MR | Zbl
[12] H. Iwaniec, Rosser’s sieve. Acta Arith. 36 (1980), 171–202. | MR | Zbl
[13] H. Iwaniec, A new form of the error term in the linear sieve. Acta Arith. 37 (1980), 307–320. | MR | Zbl
[14] H.D. Kloosterman, On the representation of numbers in the form . Acta Math. 49 (1926), 407–464.
[15] J. Liu, On Lagrange’s theorem with prime variables. Quart. J. Math. Oxford, 54 (2003), 453–462. | MR | Zbl
[16] J. Liu, M.-C. Liu, The exceptional set in the four prime squares problem. Illinois J. Math. 44 (2000), 272–293. | MR | Zbl
[17] J.Liu, T. D. Wooley, G. Yu, The quadratic Waring–Goldbach problem. J. Number Theory, 107 (2004), 298–321. | MR | Zbl
[18] V.A. Plaksin, An asymptotic formula for the number of solutions of a nonlinear equation for prime numbers. Math. USSR Izv. 18 (1982), 275–348. | Zbl
[19] P. Shields, Some applications of the sieve methods in number theory. Thesis, University of Wales 1979.
[20] D.I. Tolev, Additive problems with prime numbers of special type. Acta Arith. 96, (2000), 53–88. | MR | Zbl
[21] D.I. Tolev, Lagrange’s four squares theorem with variables of special type. Proceedings of the Session in analytic number theory and Diophantine equations, Bonner Math. Schriften, Bonn, 360 (2003). | MR | Zbl
[22] T.D. Wooley, Slim exceptional sets for sums of four squares, Proc. London Math. Soc. (3), 85 (2002), 1–21. | MR | Zbl
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