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It is proved that for a given integer N and for all but ⪡ (log N ) B prime numbers kN5/96 − ε the following is true: For any positive integers bi , i ∈ {1, 2, 3, 4, 5}, ( bi , k ) = 1 that satisfy Nb12 + b22 + b32 + b42 + b52 (mod k ), N can be written as N = p12 + p22 + p32 + p42 + p52 , where the pi , i ∈ {1, 2, 3, 4, 5} are prime numbers that satisfy pibi (mod k ).

  • Bauer, C. , A note on sums of five almost equal prime squares, Archiv der Mathematik , Vol. 69, No. 1 (1997). MR98c :11110

  • Bauer, C. , On the exceptional set for the sum of a prime and the k -th power of a prime, Studia Scientiarum Mathematicarum35 (1999), 291–330. MR2001f :11174

    Bauer C. , 'On the exceptional set for the sum of a prime and the k-th power of a prime ' (1999 ) 35 Studia Scientiarum Mathematicarum : 291 -330.

    • Search Google Scholar
  • Bauer, C. , Sums of five almost equal prime squares, Acta Mathematica Sinica21 (2005), No. 4, 833–840. MR2006i :11117

    Bauer C. , 'Sums of five almost equal prime squares ' (2005 ) 21 Acta Mathematica Sinica : 833 -840.

    • Search Google Scholar
  • Bauer, C. and Wang, Y. , On the Goldbach Conjecture in arithmetic progressions, Rocky Mountains Journal of Mathematics , Vol. 36, Number 1 (2006), 35–66. MR2007a :11143

    Wang Y. , 'On the Goldbach Conjecture in arithmetic progressions ' (2006 ) 36 Rocky Mountains Journal of Mathematics : 35 -66.

    • Search Google Scholar
  • Brünner R., Perelli, A. and Pintz, J. , The exceptional set for the sum of a prime and a square, Acta Math. Hung.53 (1989), no. 3–4, 347–365. MR2001f :11174

    Pintz J. , 'The exceptional set for the sum of a prime and a square ' (1989 ) 53 Acta Math. Hung. : 347 -365.

    • Search Google Scholar
  • Davenport, H. , Multiplicative Number Theory , 2nd ed., Springer, Berlin 1980. MR82m :10001

    Davenport H. , '', in Multiplicative Number Theory , (1980 ) -.

  • Gallagher, P. X. , A large sieve density estimate near σ = 1, Inventiones Math. , 11 (1970), 329–339. MR43 #4775

    Gallagher P. X. , 'A large sieve density estimate near σ = 1 ' (1970 ) 11 Inventiones Math. : 329 -339.

    • Search Google Scholar
  • Heath-Brown, D. R. , Prime numbers in sort intervals and a generalized Vaughan identity, Canadian J. Math.34 (1982), 1365–1377. MR84g :10075

    Heath-Brown D. R. , 'Prime numbers in sort intervals and a generalized Vaughan identity ' (1982 ) 34 Canadian J. Math. : 1365 -1377.

    • Search Google Scholar
  • Hua, L. K. , Additive theory of prime numbers, Translation of Mathematical monographs , Vol. 13, AMS, providence, RI (1965). MR33 #2614

    Hua L. K. , '', in Translation of Mathematical monographs , (1965 ) -.

  • Huxley, N. M. , Large values of Dirichlet polynomials III, Acta Arithmetica26 (1975), 435–444. MR52 #300

    Huxley N. M. , 'Large values of Dirichlet polynomials III ' (1975 ) 26 Acta Arithmetica : 435 -444.

    • Search Google Scholar
  • Liu, M. C. and Liu, J. Y. , The exceptional set in the four prime squares problem, Illinois Journal of mathematics , Vol. 44, No. 2 (Summer 2000), 273–293. MR2001i :11119

    Liu J. Y. , 'The exceptional set in the four prime squares problem ' (2000 ) 44 Illinois Journal of mathematics : 273 -293.

    • Search Google Scholar
  • Liu, J. Y. , On Lagrange’s theorem with prime variables, Quart. J. Math.54 (2003), 453–462. MR2004m :11166

    Liu J. Y. , 'On Lagrange’s theorem with prime variables ' (2003 ) 54 Quart. J. Math. : 453 -462.

    • Search Google Scholar
  • Liu, J. Y. and Zhan, T. , Hua’s theorem on prime squares in short intervals, Acta Math. Sin. (Engl. Ser.)16 (2000), 669–690. MR2001m :11169

    Zhan T. , 'Hua’s theorem on prime squares in short intervals ' (2000 ) 16 Acta Math. Sin. (Engl. Ser.) : 669 -690.

    • Search Google Scholar
  • Liu, J. Y. and Zhan, T. , An iterative method in the Goldbach-Waring problem, Chebyshevskii Sb.5 (2005), 164–179. MR2006g :11206

    Zhan T. , 'An iterative method in the Goldbach-Waring problem ' (2005 ) 5 Chebyshevskii Sb. : 164 -179.

    • Search Google Scholar
  • Liu, J. Y. and Zhan, T. , Sums of five almost equal prime squares, II, Sci. China Ser. A41 (1998), 710–722. MR99h :11115

    Zhan T. , 'Sums of five almost equal prime squares, II ' (1998 ) 41 Sci. China Ser. A : 710 -722.

    • Search Google Scholar
  • Liu, J. Y. and Zhan, T. , On sums of five almost equal prime squares, Acta Arith.77 (1996), 369–383. MR97g :11113

    Zhan T. , 'On sums of five almost equal prime squares ' (1996 ) 77 Acta Arith. : 369 -383.

  • Liu, J. Y. and Zhan, T. , Distribution of integers that are sums of three squares of primes, Acta Arith.98 (2001), 207–228. MR2002b :11139

    Zhan T. , 'Distribution of integers that are sums of three squares of primes ' (2001 ) 98 Acta Arith. : 207 -228.

    • Search Google Scholar
  • Pan, Chengdong and Pan, Chengbiao , Analytic number theory (Chinese) Beijing, Science Press, 1992.

    Pan C. , '', in Analytic number theory (Chinese) , (1992 ) -.

  • Prachar, K. , Primzahlverteilung , Springer Verlag, Berlin, Heidelberg, New York, 1978. MR81k :10060

    Prachar K. , '', in Primzahlverteilung , (1978 ) -.

  • Titchmarsh, E. C. , The Theory of the Riemann Zeta-Function , second edition, Oxford, Clarendon Press, 1986. MR88c :11049

    Titchmarsh E. C. , '', in The Theory of the Riemann Zeta-Function , (1986 ) -.

  • Vinogradov, I. M. , Estimation of certain trigonometric sums with prime variables, Izv. Acada. Nauk SSSR. Ser. Mat. , 3 , no. 4 (1939), 371–398.

    Vinogradov I. M. , 'Estimation of certain trigonometric sums with prime variables ' (1939 ) 3 Izv. Acada. Nauk SSSR. Ser. Mat. : 371 -398.

    • Search Google Scholar
  • Wang, Y. , Numbers representable by five prime squares with primes in arithmetic progressions, Acta Arithmetica90 (1999), 217–244. MR2000i :11158

    Wang Y. , 'Numbers representable by five prime squares with primes in arithmetic progressions ' (1999 ) 90 Acta Arithmetica : 217 -244.

    • Search Google Scholar
  • Zaccagini, A. , On the exceptional set for the sum of a prime and a k -th power, Mathematika39 (1992), 400–421. MR94g :11086

    Zaccagini A. , 'On the exceptional set for the sum of a prime and a k-th power ' (1992 ) 39 Mathematika : 400 -421.

    • Search Google Scholar