The following information was submitted:
Transactions: WSEAS TRANSACTIONS ON MATHEMATICS
Transactions ID Number: 53-517
Full Name: Feng Sui Liu
Position: Professor
Age: ON
Sex: Male
Address: 13-2404,Fulinyuan, Beiyuan road, BeiJing,China
Country: CHINA
Tel: 84908147
Tel prefix: 0086-10
Fax:
E-mail address: liufengs@tom.com
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Title of the Paper: On the Sophie Germain prime conjecture
Authors as they appear in the Paper: Feng Sui Liu
Email addresses of all the authors: liufengs@tom.com
Number of paper pages: 10
Abstract: By extending the operations +,¡Á on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of the second order arithmetic <P(N),N,+,¡Á,0,1,¡Ê>. We designed a recursive sieve method performed on residue classes and obtained recursive formulas of a set sequence and its subset sequence of Sophie Germain primes, both set sequences converge to the set of all Sophie Germain primes. Considering the numbers of elements of both set sequences, one is strictly monotonically increasing and another is monotonically increasing, both order topological limits exist, both are equal, we concluded that the counting function of the Sophie Germain primes approaches infinity. The cardinal function is sequentially continuous with respect to order topology, we proved that the cardinality of the set of all Sophie Germain primes is ℵ_0 using the modular arithmetical and analytic techniques on the set sequences. Further we extended this resu!
lt to attack on the Cunningham chain.
Keywords: second order arithmetic,recursive sieve method,order topology,limit of set sequences,Sophie Germain prime, Cunningham chain, Ross-Littwood paradox
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