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On the Sophie Germain Prime Conjecture

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On the Sophie Germain Prime Conjecture WSEAS TRANSACTIONS on MATHEMATICS Fengsui Liu On the Sophie Germain prime conjecture FENGSUI LIU Department of Mathematics University of NanChang NanChang China [email protected] Abstract: - By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic P(N),N,+,× ,0,1, . We designed a recursive sieve method on residue classes and obtained recursive formulas of a set sequence and its subset sequence of Sophie Germain primes, both the set sequences converge to the〈 set of all Sophie∈ 〉Germain primes. Considering the numbers of elements of this two set sequences, one is strictly monotonically increasing and the other is monotonically increasing, the order topological limits of two cardinal sequences exist and these two limits are equal, we concluded that the counting function of Sophie Germain primes approaches infinity. The cardinal function is sequentially continuous with respect to the order topology, we proved that the cardinality of the set of all Sophie Germain primes is using modular arithmetical and analytic techniques on the set sequences. Further we extended this result to attack on Twin primes, Cunningham chains and so on. ℵ0 Key Words: Second order arithmetic, Recursive sieve method, Order topology, Limit of set sequences, Sophie Germain primes, Twin primes, Cunningham chain, Ross-Littwood paradox − a 3 mod 4 1 Introduction If , then a is a Sophie Germain Primes are mysterious, in WSEAS there is a recent prime if and only if research also [7]. ≡ 2a + 1|M . Where M is a Mersenne number In number theory there are many hard a M = 2 1. conjectures about primes, the Sophie Germain prime a This theorem is astated by Euler in 1750 and conjecture is one from them. a If a, 2a + 1 are simultaneously prime, we call proved by Lagrange in− 1775. this natural number a be a Sophie Germain prime Thus the proof of the Sophie Germain prime and denote it with a predicate S(2, a). conjecture is a proof of another open problem, that The first few Sophie Germain primes are there are infinitely many Mersenne composites. 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131,…… . On the basis of heuristic prime number theory Like the twin prime conjecture, it is a conjecture and the prime theorem, Hardy and Littlewood that there are infinitely many Sophie Germain formulated the Sophie Germain prime conjecture as primes. We may express this conjecture with a very follows. simple formal sentence The number (x, S) of Sophie Germain primes S(2, a), less than or equal to a given real number x is but it is very hard to prove this sentence rigorously. approximately [2]π b a>푏 Sophie Germain∀ ∃ primes play an important role in (x, S)~2c ~ . number theory. Here are two examples. x dt 2c2 x This formula gives accurate2 predications. It is These primes were named after her when French π 2 ∫2 log t log 2t log x extremely difficult to prove this formula rigorously woman mathematician Sophie Germain (around in analytic number theory. 1825) proved that Fermat's Last Theorem holds true In 2006, Terence Tao expounded additive for such primes, this is the first general result toward patterns in primes and said: "Prime numbers have a proof of Fermat's Last Theorem. This beautiful some obvious structure. (They are mostly odd, result was extended by Legendre, and also by Denes coprime to 3, ect.) We don't know if they also have (1951), and more recently by Fee and Granville some additional exotic structure. Because of this, we (1991) [4]. have been unable to settle many questions about There is a theorem about Mersenne composites. primes."[21]. ISSN: 1109-2769 421 Issue 12, Volume 10, December 2011 WSEAS TRANSACTIONS on MATHEMATICS Fengsui Liu In 2007, G.Harman described the rapid x d mod a b development in recent decades of sieve methods and is the solution of the system of congruences i j analytic number theory, stressed again: "As is well x ≡ a mod a, known, the solution to these problems seems to be x b mod b. i well beyond all our current method."[6] If a, ≡b are coprime gcd(a,b)=1, by the Chinese j Many remarkable results have been proved using remainder≡ theorem the solution is unique and modern sieve theory. Because the parity problem, a computable. formidable obstacle, the traditional sieve theory is Except extending +, × to finite sets of natural unable to settle those conjecture [9] [21]. One needs numbers, we continue the traditional interpretation to recast the sieve method. of symbols +,×, , then we founded a now model of Based on the modern progress of model theory, the arithmetic of natural numbers by a two-sorted set theory, general topology and recursion theory we logic ∈ try to tackle those problems with a new sieve P(N), N, +, ×, 0,1 , . method. Where N is the set of natural numbers and P(N) is In an exotic realm of a second order arithmetic the power set〈 of N. ∈〉 P(N), we introduce set sequences of natural numbers, In contrast to the usual first order arithmetical and use the Chinese remainder theorem to recast or model refine Eratosthene's sieve method, then we can N, +,×, 0, 1 , capture some enough usable secret structures about we call this new model be a second order prime sets and use limits of set sequences to directly arithmetical model〈 or algebraic〉 structure and denote determine the set of all Sophie Germain primes and it with P(N). its cardinality. Mathematicians assume that N, +,×, 0, 1 is the standard model of Peano theory PA, similarly, we {a: S(2, a)} = limA T = lim , assume that P(N),N,+,×,0,1, 〈 is the standard〉 |{a: S(2, a)}| = |limA |=|lim′ T | = ′lim|T | = . i i model of the theory of the second order arithmetic ′ ′ ′ PA ZF, which〈 is a joint theory∈ 〉of PA and ZF, in i i i ℵ0 By this formulation, it is comparatively easy to other words the P(N) not only is a model of Peano prove the Sophie Germain prime conjecture, theory∪ PA but also is a model of set theory ZF. because this formula itself has discerned a logical As a model of set theory ZF, the natural numbers structure of the set of all Sophie Germain primes. are atoms or urelements, objects that have no The parity problem in the traditional sieve theory element. We discuss the sets of natural numbers and would naturally be circumvented. sets of sets of natural numbers. In the second order formal system P(N), we 2 A formal system formalize natural numbers and sets thereof as First of all we define several operations on finite individuals, terms or points. A determined set not sets of natural numbers. only contains its all elements but also includes all Let information of the distribution of its elements, A = a , a , … , a , … , a , especially the number of elements in this set. B = b , b , … , b , … , b The second order language +,×, 0, 1, has 1 2 i n be arbitrary finite sets〈 of natural numbers,〉 we define stronger expressive power. The second order model 1 2 j m A + B = a + b〈, a + b , … , a + b〉, … , a + has richer mathematical structures.〈 The ∈sec〉 ond b , a + b , order theory is more powerful and flexible. 1 1 2 1 i j n−1 〈 Except inheriting usual theorems of the first AB = a b , a b , … , a b , … , a b , a b . m n m For the empty set , 〉let order arithmetic, in P(N) we may introduce a new 1 1 2 1 i j n−1 m n m 〈 + A = , 〉 sort of mathematical structures and to prove some hard conjectures about primes. The first order A = .∅ arithmetic N has no well formed formula to Let A\B be∅ the set subtraction.∅ represent such new structures and to prove those Define the∅ solution∅ of the system of congruences conjectures about primes. X A = a , a , … , a , … , a mod a , In particular we introduce a new sort of X B = b , b , … , b , … , b , mod b ≡ 〈 1 2 i m〉 arithmetic functions to be f: N P(N) ≡ 〈 1 2 j m 〉 , X D = d , d , … , d , … , d , d mod ab. Where → ≡ 〈 11 21 i j n−1 m n m〉 ISSN: 1109-2769 422 Issue 12, Volume 10, December 2011 WSEAS TRANSACTIONS on MATHEMATICS Fengsui Liu which is a set valued function defined on the set N, It reveals a fundamental rule of the distribution of i.e., set sequences of natural numbers, to obtain a primes. Now the primes appear in a highly regular recursive sieve method on the residue classes. pattern. We may use the recursive sieve method to look Note, the traditional sieve theory itself is unable for a string of logical reasoning that demonstrates to prove Euclid's theorem [21]. the truth of the Sophie Germain prime conjecture is We do not discuss further this formal system in built into the structures of prime sets in the view from logic [14]. framework of recursion theory and general topology, rather than to look for good approximations or non- 3 A recursive formula and its simple trivial lower bounds of the counting function (x, S). consequences "A well chosen notation can contribute to making An ancient Greek mathematician Eratosthenes mathematical reasoning itself easier, or even πpurely created a sieve method for fining the primes in a mechanical."[11]. given range. As a simple example of the recursive sieve Based on the inclusion-exclusion principle, method, deleting the congruence class 0 mod p Lagendre used the sieve method of Eratosthenes to from the set of all natural numbers successively, i.e., i estimate the size of sifted sets of integers in a given all numbers a such that the least prime factor of a is range.
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