By Vinogradov I. M.

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8 1 - -4LT^<2"+1'"<-1Corresponding results of twin-primes pro' --ms have also been obtained. , 1957, On Sieve Methods and Some of the Related Problems, Science Record, Academia Sinica, New Ser. I, 1, 9 — 12. , 1957, On Sieve Methods and Their Some Applications, Science Record, Academia Sinica, New Ser. I, 3. 1—6. , 1956. On the Representation of Large Even Integer as a Sum of a Product of a t most 3 Primes and a Product of a t most 4 Primes, Acta Math. Sinica, 3. 500-513. [4] ByxniTa6, A. , 1940.

1. M. N. KL. 3, 1-36. , 1942, Zur Viggo Brun'schen Siebmethode, I Norske Vid. Selsk. Fork. Trondhjem, 39, 145-148. , 1953, Neue Abschatzungen auf Grund der Viggo Brunschen Siebmethode, Tolfte Skandinaviska Matematikerkongressen, Lund, 160-168. , 1956, On the Representation of Large Even Integer as a Sum of a Product of at Most 3 Primes and a Product of at Most 4 Primes, A eta Mathemalica Simca, 6:3, 500-513. , 1956, On the Representation of Large Even Integer as a Sum of a Prime and a Product of at Most 4 Primes, Ada Malhematica Sinica, 6:4, 565-5S2.

29 SCIENCE RECORD New Ser. Vol. I, No. 3, 1957 MATHEMATICS ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* WANG YUAN (£ 'It) Institute of Mathematics, Academia Sinica {Communicated by Prof. Hua, L. , Member of Academia Sinica) In 1919, V. BruntI] first proved the following result: For sufficiently large x, there exists, between x — x2 and x, an integer of at most 11 primes factors. Brun's method and his results were improved by several Mathematicians. The best result before my work was due to P.

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