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### On additive problems with prime numbers of special type

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Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. It is proved that for almost all sufficiently large integers n, satisfying n is identical with 0 or 1 (mod 3), the equation n = p1+p2/2 +p2/3 has a solution in primes p1, p2, p3 such that p1+2 = P6, p2+2 = P5, p3+2 = P5. It is also proved that for every suffciently large integer M is identical with 0 or 2 (mod 3), the equation M = p1+p2/2+p2/3+p2/4+p2/5 has a solution in primes p1, ź ź ź , p5 such that p1+2 = P6, p2+2 = P5, p3+2 = P5, p4+2 = P2, p5+2 = P'/2.
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271--287
Opis fizyczny
Bibliogr. 17 poz.
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autor
autor
• Department of Statistics and Mathematics Shandong Finance Institute Jinan, 250014, P.R. China, mengxm@beelink.com
Bibliografia
• [1] J. Brudern and E. Fouvry, Lagrange's four squares theorem with almost prime variables, J. Reine Angew. Math. 454 (1994), 59-96.
• [2] J.-R. Chen, On the representation of a large even intger as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176.
• [3] H. Davenport, Multiplicative Number Theory (revised by H. Montgomery), 2nd ed., Springer, 1980.
• [4] R. R. Hall and G. Tenenbaum, Divisors, Cambridge Univ. Press, 1988.
• [5] G. H. Hardy and J. E. Littlewood, Some problems of partitio numerorum III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
• [6] L.-K. Hua, Introduction to Number Theory, Springer, 1982.
• [7] L.-K. Hua, Some results in the additive prime number theory, Quart. J. Math. 9 (1938), 68-80.
• [8] H. Iwaniec, Rosser's sieve, Acta Arith. 36 (1980), 171-202.
• [9] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith. 37 (1980), 307-320.
• [10] Y. Linnik, An asymptotic formula in an additive problem of Hardy and Littlewood, Izv. Akad. Nauk SSSR. Ser. Mat., 24 (1960), 629-706.
• [11] H. Mikawa, On the sum of three squares of primes, in: Analytic Number Theory, London, Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, 1997.
• [12] H. Mikawa, On exponential sums over primes in arithmetic progressions, Tsukuba J. Math. 24 (2000), 351-360.
• [13] T. P. Peneva and D. I. Tolev, An additive problem with primes and almost-primes, Acta Arith. 83 (1998), 155-169.
• [14] D. I. Tolev, Arithmetic progrssions of prime-almost-prime twins, Acta Arith. 88 (1999), 67-98.
• [15] D. I. Tolev, Additive problems with prime numbers of special type, Acta Arith. XCVI (2000), 53-88.
• [16] M. Q. Wang, On sums of a prime and two squares of prime, Acta Math. Sinica, 147 (2004).
• [17] M. Q. Wang, X. M. Meng, The exceptional set in the two prime squares and a prime problem, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1329-1342.
Typ dokumentu
Bibliografia