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 − 1 Introduction If a 3 mod 4 , 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].

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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〉

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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. A main difficult is the accumulation of error p , instead the multiples of p in a given range, terms leaving the reduced residue systemT mod m , i i Taking a partial summation from the formula of we obtain an exact formula producing all primes[16]: the inclusion-exclusion principle, Brun estimated T = 1 , i+1 i+1 the size of sifted sets for almost primes, products of p = 3, 1 at most k primes. T = (T 〈+〉 m 0,1,2, … , p 1 )\ D , 1 According to carefully chosen weight functions, p = U(T ). (1) i+1 i i i i Selberg given a better estimate for the almost primes. Where the U(T 〈) is〉〈 a projective function− 〉 i+1 i+1 Some partial successes of the traditional sieve U( t , t , … , t ) = t , i+1 theory include: Brun’s theorem the sum of the p = 2, 1 2 n 2 reciprocals of twin primes converges, Chen’s m 〈 = p ,〉 0 theorem there are infinitely many primes p such that and i i+1 0 j p + 2 is the product of at most two primes[3], and X D =∏ p T mod m the fundamental lemma of sieve theory. is the solution of the system of congruences One of original purposes of the traditional sieve i i i i+1 X ≡ 0 mod〈 〉p , theory was to settle prime patterns of various forms, X T mod m . example the twin primes or the Sophie Germain i The first≡ few〈 〉 terms of this formula are: primes. i i T ≡= 1 , It has proven that the traditional sieve theory is p = 3, unable to provide non-trivial lower bounds on the 1 T = 〈1,3,5〉 \ 3 = 1,5 , size of prime patterns. Also, any upper bounds must 1 p = 5, be off from the truth by a factor of 2 or more. This is 2 T = 〈1,5,7,11〉 〈,13〉 ,17〈,19,23〉 ,25,29 \ 5,25 the parity problem[21]. 2 = 1,7,11,13,17,19,23,29 It seems that one needs basically to recast the 3 P = 7〈 〉 〈 〉 traditional sieve theory to get around the parity T =〈 1,7,11,13,17,…,199,203〉 ,209 \ problem. 3 7,49,77,91,119,133,161,203 Now we recast the sieve method of Eratosthenes 4 = 〈1,11,13,17,…,197,199,209 〉 using the Chinese remainder theorem. We try to p =〈 11, 〉 understand what is the sieve of Eratosthenes in view It is easy〈 to prove this formula by 〉means of from modern mathematics and logic, and then to mathematical4 induction: the least number except 1 prove Sophie Germain prime conjecture. in the reduced residue system T is the prime p . Let p be the i-th prime, p = 2. One plugs i into this formula, this formula will For any prime p > 2， we consider the i+1 i+1 i 0 produce the i-th prime. This primitive recursive congruence classes i formula actually exhibits infinitely many primes by B = {a: a 0 mod p 2a + 1 0 mod p } +, ×, it provides a constructive proof of Euclid's 0, (p 1) mod p . theorem using the new sieve method[19]. i ≡ i ∨ ≡ i Let 1 ≡ 〈 2 i − 〉 i

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m = p . 2) Using the recursive formula (2), we easily From the set of all oddi numbers compute Sophie Germain primes, in fact, we had i+1 0 j X 1 mod∏ 2 computed out the first few Sophie Germain primes we successively delete the congruence classes 3,5,11,23,29,41,53,83,89,113,131,173,179 ,191,…. X ≡ 〈 B〉 mod p , i.e., all numbers a such that the least prime factor of 3) If a p is a Sophie Germain prime, then the ≡ i i natural number a belongs to the congruence class T a or 2a+1 is p , and obtain the congruence classes i X T mod m a ≥T mod m . i i such that if a X then a and 2a+1 do not contain i+1 i+1 i i any prime ≡p p as a factor. As an∈ algorithm for Sophie Germain primes, like So that a recursive∈ formula of T which is the the prime formula, one may further observe a fact j i T T set of the least≤ nonnegative representatives of the that the least number in the set min is a Sophie i+1 residue class mod m is as follows: Germain prime for all i a = min T S(2, a), i i T = 1 , i+1 then intuitively our sieve method provides a T = (T + m 0,1,2, … , p 1 )\D . (2) i 1 → Where 〈 〉 conclusion that there are infinitely many Sophie i+1 i i i i Germain primes, but it is not easy to directly prove X D mod〈 m〉〈 − 〉 this fact. We do not discuss this problem in detail. is the solution of the system of congruences i i+1 X ≡ T mod m X B mod p . 4 A Sophie Germain prime theorem i i The number≡ of elements of the set T is The sifting process on residue classes products a i i |T ≡| = p 2 . (3) recursive set sequence T , we refine this recursive i+1 set sequence to determine the set of all Sophie The first few termsi of those sets T are i i+1 1 j Germain primes and its cardinality using analytic T = 1 , ∏ � − � i techniques. T = ( 1 + 0,2,4 ) \ 1,3 = 5 , 1 T = 〈( 5〉 + 0,6,12,18,24 ) \ 5,17 , 2 Let B denote the set of all non-Sophie Germain = 11〈 ,〉23,39〈 , 〉 〈 〉 〈 〉 3 primes in′ the congruence classe B T = (〈 11〉 ,23〈 ,29 + 0,30〉,…,〈150,180〉 ) i B = {a: (a 0 mod p 2a + 1 0 modp ) 〈\ 59,101,〉119,143,161,203 i 4 ′ a > p } = 11〈 ,23,29,41〉 ,53〈 ,…,173,179 ,191〉 ,209 . i i i We delete ≡the set B and∨ save the≡ natural number∧ It is easy to〈 prove the formulas (2), 〉(3) by means i a as a survivor if a is a ′Sophie Germain prime. of mathematical〈 induction. 〉 A i Like Eratosthene's sieve, the number 1 is hidden, Let be the set of all Sophie Germain primes less than p which does not enter the filtration, all odd numbers i A = {a: a < p S(2, a)}. a > 1 are filtrated. i We adjust the set T to be We delete the numbers a in the congruence i i T = A T ∧ (5) classes i X B mod p Except′ saving all Sophie Germain primes a < i i ∪ i under the divisible condition p as survivors in the set T , two set sequences T i i and T are same. ′ ′ {a: (a ≡ 0 mod p 2a + 1 0 modp ) a p }, i i i Let |A | be the number of all Sophie Germain then except a = p itself may be a Sophie Germain i ≡ i ∨ ≡ i ∧ ≥ i primes less than p , then the number of elements of prime, other numbers a all are not Sophie Germain i i the set T is prime. i |′T | = |A | + |T |. (6) Now we list some simple consequences from the i T Let A be′ the Sophie Germain prime subset of recursive formula , their proof is easy. i i i the set T′ i T i T A′ = {a: a T S(2, a)} (7) 1) Let min is the least number in the set , then i a criterion of Sophie Germain prime is ′ ′ i i i i S(2, a) a = min T = p 3. (4) 4.1 An informal∈ argument∧ This criterion recursively enumerates all Sophie Now we describe how the sifting process T , A i i Germain primes.↔ ≥ approaches the infinite set of all Sophie Germain′ ′ primes in the framework of an order topology. i i Obtained the set T from the formula (5), let a T , if ′ i ′ ∈ i

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2a + 1 < p , said that one performed a supertask [13], and have then the number a is a Sophie Germain2 prime, which obtained infinitely many natural numbers a such that belongs to all T for r > i and willi never be deleted. a and 2a+1 do not contain any prime as a factor Obtained the′ set T from the formula (5), let except itself, these infinitely many natural numbers r a T , if ′ a exactly constitute the set of all Sophie Germain i ′ 2a + 1 p , primes. i then∈ a is a good candidate for 2the Sophie Germain Our sieve method can exactly determine that the prime, a and 2a+1 do not contain≥ i any one of the first numbers of elements of the set sequence T is strictly monotonically increasing, we need ′not i -1 primes as a factor. In this case, if a is a Sophie i Germain prime, then a belongs to all T for r > i, estimate the lower bounds or upper bounds of the ( ) otherwise the a is an "error term", there ′is a prime counting function x, S through the almost primes. p such that r We get around the parity problem. π p | a ∨ p |2a+1, Our proof needs not the Cramer random model k a does not belong to any T for r > k. or The Riemann Hypothesis. k k This is an informal argument, which shows the Our sieve method itself ′will delete all error terms and needs not estimate ther number of error terms. Sophie Germain prime conjecture is true. Each non- Sophie Germain prime is deleted We give a formal proof in the sense of logic exactly once, our sieve method needs not the using analytic techniques in an order topological inclusion-exclusion principle. There is no space. accumulation of error terms. As i goes to infinity we delete more and more non-Sophie Germain primes, exhibit more and more 4.2 A formal argument Sophie Germain primes or candidates of Sophie Germain primes in the set T , the set sequence T Lemma 1 The set sequences , converge to the gets as close as we want to the′ infinite set of Sophie set of all Sophie Germain primes.′ ′ i 푖 푖 Germain primes, the set sequencesi T , A get as 푇 퐴 close as we want to the infinite set {a:S(2,a)}′ ′ of all Proof: First of all we quote a definition of the set Sophie Germain primes. i i theoretic limit of a set sequence from the textbook If i is extremely large, example ''Set Theory, with an introduction to descriptive set theory'' by K.Kuratowski and A.Mostowski [15]. i=c=10 , 1000 Let F denote a set sequence, we define theoretically we can construct10 a set by the formula limsup F = F , (5), which has approximated to the set {a:S(2,a)}of n liminf F = ∞ ∞ F , all Sophie Germain primes. n=∞ n n=0 i=0 n+i If ∩∞ ∪∞ By the prime theorem, the set T has exactly n=∞ n n=0 n+i limsup F =∪ liminf∩i=0 F , exhibited all Sophie Germain primes ′ c we say that the set sequence F converges to the n=∞ n n=∞ n limit a < 1/2p ~10 × 10 × 2.3 × 10 n 1000 lim F = limsup F = liminf F . 2 1000× 10 10 1000 c Let 1000 n n=∞ n n=∞ n The set T has 10 T = {a: S (2, a)} ′ |T | > c p 2 be the set of all Sophie Germain primes, the sifted 101000 e elements′ a such10 that a and 2a+1 do not contain any set, which is the end result by deleting all non- i ∏1 � j − � one of the first 10 -1 primes as a factor except Sophie Germain primes from the set of all odd 1000 numbers. itself, in other words,10 if a or 2a+1 has prime factors except itself, they are large than Let X be the whole congruence class mod m with the representatives T , 2.3 × 10 × 10 . i i 1000 X T mod m In the approximate1000 10 sense the set T may be i Let regarded as a set of Sophie Germain primes′ and the i i i c X =≡ A X number of elements of this set may be regarded as Then ′ an infinity. Against our daily standard the set T is i i i X ） X ） … ）∪X ） … …. an infinite set of Sophie Germain primes. ′ , c according to the′ definition′ of the′ set theoretic limit Ultimately, as the limit of the set sequences T , 1 2 i A we have deleted all non- Sophie Germain prime′ of a set sequence we have i sets′ B successively, philosophers of mathematics lim X = X = T . i ′ ′ ′ e i i ∩ i

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Obviously hence when we endow the well ordered set (9) with A A … A … …., an order topology, the well ordered set (10) will be we have also′ ′ ′ automatically endowed an order topology. 1 2 lim ∁A = ∁ A ∁=i T∁ . We had computed out patterns of the first few Obviously, ′ ′ Sophie Germain primes e A Ti ∪X i a=3,5,11,23,29,41,53,83,89,113,131,173,179 ,191, It is easy to′ prove′ ′ hance the numbers of elements of both the well limsupi ∁ Ti ∁ limsupi X = lim X , ordered sets (9),(10) are more than one. We can liminf T ′） liminf A ′= lim A ′ safely endow the well ordered set (9) with a i ∁ i i naturally compatible order topology, such that the liminf T′ limsup T′. ′ i i i well ordered set (10) is automatically endowed an Thus ′ ′ i i order topology. lim T ∁= T . Obviously, for every neighborhood (c, T ] of T In the sense′ of the set theoretic limit we have i e there is a natural number i , for all i > i , proved that both the set sequences T , A converge e e T (c, T ]， to their common limit point T ′ ′ 0 0 i i A′ (c, T ] lim T = lim A = T . (8) i e e thus both the set′ ∈ sequences T , A converge to their We want to′ determine′ the cardinality of the set i e i i e common limit point∈ T . ′ ′ of all Sophie Germain primes by convergence and i i lim T = lim A = T . (11) continuity of the cardinal function, where e In the sense′ of the ′order topological limit we continuous function is a map preserving the i i e topological structure between topological spaces. have proved also that both the set sequences T , A converge to the identical limit point T . ′ ′ The set theoretic limit itself does not involve any i i □ topology, it is unable to determine the cardinality of e the set of all Sophie Germain primes directly. We If the sifted set T is empty T = under some need to reveal an order topological structure of the sifting conditions, the sequence (10) contains only e e equality (8). one element , it has no linear order∅ structure, we Now we quote a definition of the order topology can not endow the sequence (10) with an order from the textbook ''Topology'' by J.R. Munkres [12]. topology by ∅the definition. Otherwise we will fall The order topology is a topology on the non- into a Ross-Littwood paradox. In the last of this empty linear order set, which contains more than section we will discuss this paradox in detail. one element, their open sets are the sets that are the unions of open intervals (c, d) and half-open Lemma 2 The cardinal sequences | |, | | intervals [c , d), (c, d ], where c is the smallest converge to the smallest infinite cardinality ′ . ′ 푖 푖 element and d is the largest element of the linear 푇 퐴 0 order set. The0 empty 0and the set0s with a single Like Euler used the product formula ℵfor the element have no0 linear order structure, they have no Riemann zeta function order topology. 1 1 = 1 According on the sifting procedure (5), the p p −1 particular set theoretic limit above to reprove Euclid's� s theorem� � that− theres� are infinitely lim T = lim A = T . many primes [5], we take the cardinality on two has equipped a′ natural ′ order structure. In other i i e sides of the equality (11) and consider the limits of words, both the set sequences T ,A and their limit cardinal sequences |T |, |A | as point T construct non-empty well′ ordered′ sets with T , A T′ ′ i i i i the order type + 1 to determine the′ cardi′ nality of the sifted set T . e i i e X: T , T ,… ,T , … … ; T . (9) About the limit →of a function, in general topology ω e ′ ′ ′ it is easy to prove: X: A1 ,A2,…,Ai ,……; Te . (10) We try to endow′ ′ the′ well ordered sets (9), (10) "If the space X satisfies the first axiom of 1 2 i e with an order topology. countability at the point x and the space Y is From the formula (7), A is the Sophie Germain Hausdorff, then for the existence of the limit 0 prime subset of T ′ lim f(x) i of a mapping A T′ , f: E Y, E X , i x→x0 ′ ′ it is necessary and sufficient that for any sequence i ∁ i x → E, ∁ n=1,2,3,……,

n ∈

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such that lim x = x , the limit lim f(x ) Sophie Germain primes and tries to prove the exists. If this condition holds, the limit Sophie Germain prime conjecture by proving that lim n→∞f(n) n 0 n→∞ n lim (x, S) is infinite. Unfortunately, one has never does not depend on the choice of the sequence x , proved that lim (x, S) is infinite or not by using all n→∞ and the common value of these limits is the limit of our πcurrent method. n f at x ." [18]. We regard |A π|, |T | as the real functions. In other words for every sequence satisfying this Obviously ′ ′ 0 condition x , the limits of functions limi (x,i S)= lim (m , S), lim f(n) now we easily obtain n i are equal. lim (x, S)π = lim π(m , S) = lim| | = n→∞ lim| | = . (17)′ i 푖 Proof: From the formula (6) we have Then we directlyπ′ prove thatπ lim (x, S) is infinite퐴 | T | |T |. using the recursive푇푖 ∞ sieve method. In the usual sense From the′ formula (3) we obtain that the cardinal we have proved the Sophie πGermain prime i i sequence |T≥| of the sets T , is strictly monotonically conjecture. increasing ′ ′ □ i i |T | < |T |. From topology we know that the value |T | of Thus ′ ′ counting functions at T is irrelevant to the i i+1 e lim | T | = |T | = . definition of the limits of cardinal functions e The ′cardinal′ sequence′ |T | and its limit point lim |T |, lim| |, lim (x, S). We need to prove the Ti →Te i ∪ i ℵ0 construct a non-empty well′ ordered set with the continuity′ of the′ cardinal function at the point T , i i 푖 order type + 1 then obtain 퐴 π 0 e ℵ Y : |T |,|T |,…, |T |,……; . (12) |T |= . ω Like the lemma′ ′ 4.1, endowing′ this well ordered e 0 1 2 i 0 ℵ set with an order topology, in the senseℵ of the order Theorem 1 The set of all Sophie Germain primes is topological limit we obtain that the cardinal an infinite set. sequence|T | of the sets T converges to the smallest infinite cardinality′ as ′T T Proof: Let f: X Y be the cardinal function from i i lim | T | = |′T | = . (13) the order topological space X to the order ℵ0 i → e Obviously′ the cardinal′ sequence′ |A | of the sets A topological space →Y, Ti →Te i ∪ i ℵ0 is monotonically increasing ′ ′ f (T) = | T |, i i | A | | A |. X: T , T , … , T , … … ; T , Thus ′ ′ i i+1 ： |T′|, |T′ |, … , |T′ |, … … ; lim | A≤| = |A |. Y 1 2 i e . It is easy′ to che′ ek: for′ every open set [|T |, ′ ′ ′ 1 2 i 0 The limiti eof the cardinal sequence |A | exists, ℵ A →T i ∪ i T ′ although we do not know whether |A | is′ infinite |d|),(|c|, |d|), (|c|, ] in Y, their preimages [ ,d), 1 i ′ or not. ′ (c,d), (c, ] are 0open sets also in X, thus the1 i ℵ For the Sophie Germain primes we∪ know cardinal function f (T)= |T| is continuous with 0 T , respect toℵ the above order topology. the cardinal sequence |A | and its limit point |A | Both the order topological spaces are first e construct a non-empty≠ ∅ well′ ordered set ′ countable, thus the cardinal function |T| is i i Y : |A |, |A |,…, |A |, … … ; |A | , ∪ (14) sequentially continuous. By the usual topological theorem, the cardinal function |T| preserves limit, may be endowed′ with′ an ′order topology′ and in the i i i i | = lim| | sense of the order topological limit ∪we have | lim lim | A | = |A |. (15) Order topological′ spaces′ are Hausdorff spaces, in Hausdorff spaces푇푖 the limit푇푖 points of the set sequence The order ′topological′ space′ is first countable and Ai →Te i i T and the cardinal sequence |T | are unique Hausdorff. Like the Euler∪ product formula, the provided′ that it exists. ′ limits of the cardinal sequences of two sides of the i The lemmas 1), lemmas 2) have provedi that there equality (11) exist, two limits are equal exist the order topological limits lim T , lim| | lim| | = lim| |= . (16) and the condition for existence of both the′ limits is′ In the traditional′ sieve′ theory or analytic number sufficient. i 푇푖 퐴푖 푇푖 ℵ0 (x, S) theory, one uses the counting function , From the lemmas 2) we have which is a real function, to model the number of π

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| lim | = lim| | = We ignore physically plausible space-time From the lemmas′ 1) we′ obtain that the set of all continuity conditions and consider what is the limit 0 Sophie Germain푇푖 primes푇 is푖 an infiniteℵ set lim T of the set sequence T . A similar example concerning the distribution of i i |{a:S(2,a}|=| lim | = | lim | = lim| | = primes is the primes in the reducible polynomial . ′ ′ ′ (18) a 1, a > 2. 푖 푖 푖 퐴 푇 푇 Under the2 sifting condition In otherℵ0 words, we have rigorously proved the B −= 1, 1 mod p . Sophie Germain prime theorem that there are One seeks for the lim T , where the number a = 2, i i infinitely many Sophie Germain primes. a - 1 = 1 does〈 not− enter〉 the filtration. Similarly, we can prove the continuity of the Their set theoretic limiti is empty counting functions |A | or (x, S) at the point T . lim T = . ′ □ Their cardinal limit itself is infinite i e i We known, one hardlyπ handles the counting lim |T | =∅ A (x, S) lim | | At first sight this is a contradiction. functions | |, , and their limits i 0 lim (x, S) directly,′ but easily handles the function′ Through thoroughℵ investigation, the condition for i 푖 |T |, and its limitπ lim | |. Taking the cardinality퐴 the existence of the cardinal limit lim |T | is not ′ π ′ sufficient with respect to the order topology, on two sides of the equality (11), we had assembled i ani order topological structure푇푖 of the set of all Sophie because the cardinal sequence A = has no order Germain primes piecemeal by the sifting process T , topological limit. There is no continuity,′ |T | and i ∅ A , the difficulty is reduced to a proof of the′ lim |T | are irrelevant, there is no contradiction. i e ′ One can not regard lim T as a cardinality of the existence about the limit of the cardinal sequence i Ai T empty. In view from pour mathematics, the order | |. For Sophie Germain primes, , the order i topology′ (9) induces a natural order topology (10) topology distinguishes non-empty sifted sets from e oni the set T , corresponding with this,≠ (9) ∅ induces a the empty sifted set and provides a formal solution natural order topology on the cardinal sequence (14), of the Ross-Littwood paradox. it is easy toe prove the existence (15) of the limit If we relax the restrictive condition and seek for lim | |, then we discern the mystery about primes a natural number a such that a little′ by the recursive sieve method. a 1, a > 2, 퐴푖 is a product2 of at most two primes, an almost prime, 4. 3 About the Ross-Littwood paradox or a product− of exactly two primes, a semiprime, Like Euler product formula, using the order then such natural numbers a have patterns topological limits we need to be a little careful about a=4,6,12,18,30,42,60,72,102,108,138,150,…, whether two sides of a formula converge [5]. a 1= 3× 5, 5× 7,11 × 13,17 × 19,29 × 31, …, Only if there is not any survivor as a pattern and2 the set lim T = T of all such numbers a is infinite.− ′ under some sifting conditions, the set theoretical i e limit (8) This result is similar to the famous results of lim T = lim A = T = . J.R.Chen [3] or H.Iwaniec [8]. Obviously |T′ | = 0. Nothing′ may be proved by the It is interesting that this is a proof of the twin e above orderi topologicali reasoning,∅ because empty prime conjecture via the polynomial a 1. set sequence e(10) has no any order topology, we can 2 not endow the set sequence (9) with an order 5 A Cunningham chain theorem− topology also, needless say continuous or not. We may uniformly extend the above sieve method In the informal argument, like lim (x, S), one and reasoning to a wide variety of additive patterns extends the reasoning paradigm and regards lim| | in primes. as a cardinality of the empty, then falls πinto a Ross′- Let B be the solution of the congruence Littwood paradox [10] [17] [20], today this paradox푇푖 cx + d o mod p , i is an argumentative problem also. we obtain a different proof of Dirichlet's theorem i 1953. J.E. Littlewood described the following that there are ≡infinitely many primes of the form paradox about infinity. cx + d, gcd (c, d) = 1. Balls numbered Let B be the solution of the congruence T = i + 1, i + 2, … ,10i x + 1 0 mod p , i are put into a urn. How many balls are in the urn at we prove2 that the number of primes of the form x i i the end as i 〈 ? 〉 + 1 is infimite. ≡ 2

→ ∞

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Let Thus we only discuss the admissible Cunningham X B = 1, 2 mod p , chains. We prove the twin prime conjecture. Similarly we Like above proof, from the set of all odd may prove≡ thei k-〈tuple− prime〉 conjecture.i numbers Let X 1 mod 2, X B = 1, mod p , we delete the congruence classes pi−1 X ≡ 〈 B〉 = b , b , … , b , … , b mod p , repeat the above reasoning, we prove that there exist ≡ i 〈 4 〉 i successively, and obtain the recursive formula T . infinitely many primes p such that 4p+1 is prime. i 1 2 r k i The number≡ of elements〈 of the set T is 〉 It is easy to prove that if there exist infinitely i (19) |T | = p w many primes p such that 4p+1 is prime, then 2 is a i primitive root modulo q for infinitely many primes If p > k+1, w ki−1, then the | T | is strictly i ∏1 j j monotonically increasing �for −i > .� q = 4p + 1. j j i Thus we prove that Artin’s conjecture, an open We have known≤ that there is a pattern a of the problem, is true at 2. Cunningham chain, thus the sifted푗 set is not empty. 1 One naturally extends Sophie Germain primes to We refine this recursive set sequence T and Cunningham chains [1]. obtain two set sequences T , . i A Cunningham chain of length k is a finite set of Further we prove that ′ both′ the set sequences primes T , converge to the siftedi 퐴푖 set T , the cardinal a , a , … , a , functions′ ′ |T |, | | are continuous with respect to the i 푖 e Where order퐴 topology.′ ′ 1 2 k a = 2a +1, r=1,2,…,k 1, Thus we iproved퐴푖 the Cunningham chain theorem. each which is twice the proceeding one plus one. |{a:S(k,a)}| = | lim | =|limT |= lim | T |= . r+1 r For example: − ′ ′ ′ (20) 푖 i i 0 2, 5, 11, 23, 47 ; This completes the proof..퐴 ℵ □ 89, 179, 359, 719, 1439, 2879. Like the k-tuple prime conjecture, it is We do not know whether there exists a conjectured that there are infinitely many Cunningham chain of length k > 17, if we find a Cunningham chains of length k. Cunningham chain of length k > 17 and it is Extending the above proof we obtain a theorem admissible then its number is infinite. following. According our sieve method, many additive patterns in primes have the same structure of set Theorem 2 The number of admissible Cunningham theory, structure of order topology and recursive chains known of length k is infinite. structure, although they have different arithmetical structures. The methods proving infinity or not of Proof: For any Cunningham chain various sifted sets T all are same. In this short a , a , … , a , paper we do not discuss those problems in detail. let e 1 2 k B =〈 b , b , … , b〉, … , b , where b is the solution of the congruence i 1 2 r k Acknowledgements: I’d like to thank Professor 2 x + 2 〈 1 0 mod p , 〉 r Wang ShiQiang of Beijing Normal University, who r = 1, 2,r ..., k. i gave me directions on model theory. My thanks also Let w be −the number≡ of congruence classes in go to Professor Mo ShaoKui of NanJing University, B mod p , with his guidance on recursion theory, Thanks i If there is a prime p , the Cunningham chain Professor Fu WanTao and Lai YingHua for their i i covers all congruence classes mod p excellent advice. i B 0,1,2, … , k 1 mod p , i we say that this Cunningham chain is inadmissible. References: i i For ≡ an〈 inadmissibl− e〉 Cunningham chain, [1] A. Cunnningham, On hyper-even numbers and obviously, if n > i, then the set T is empty, the on Fermat's numbers, Proc. Lond. Math. Soc., recursive sieve method itself proves that there is no series 2, 5, 1907, pp237--274. n any natural number c > p such that [2] Chris K. Caldwell, An amazing prime heuristic, a + c, a + c, … , a + c 2000. http://www.utm.edu./caldweii// i are simultaneously prime. [3] J. R. Chen, On the Representation of a Large 〈 1 2 k 〉 Even Integer as the Sum of a prim and the

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Product of at Most Two Primes, Kexue Tongbao [19] P. Ribenboim, The new book of prime number Vol 17, 1966, pp385—386. records, 3rd edition, Springer-Verlag, New York, J. R Chen, On the Representation of a Large 1996. Even Integer as the Sum of a prime and the [20] S.Ross, A First Course in Probability, third Product of at Most Two Primes, I.Sci.Sinica Vol edition, New York and London, Macmillan, 1988. 16, 1973, pp157—176. [21] Terence Tao, Open question: The parity J. R. Chen, On the Representation of a Large problem in sieve theory, Even Interger as the Sum of a prime and the http://terrytao.wordpress.com/2007/06/05/open- Product of at Most Two Primes, II.Sci.Sinica Vol question-the-parity -problem-in-sieve-theory/ 16, 1978, pp412—430. Terence Tao, Long arithmetic progressions in [4] G. Fee and A. Granville, The prime factors of the primes, Auatrarian mathematical sosiety Wendt's binomial circulant determinant, Math. meeting, 2006. Comp., Vol 57:196, 1991, pp 839—848. [5] A. Granville, Analytic number theory, http://www.dms.umontreal.ca/%7Eandrew/PDF/P rinceComp.pdf [6] G. Harman, Prime-detecting sieve, Princeton university press, 2007 [7] E. Hibbs, Applying the new primer on prime numbers, WSEAS Transactions on mathematics, Vol. 9, 2010, pp.234-243. [8] H. Iwanies, Almost-primes represented by quadratic polynomials, Ivent.Math. Vol 47, 1978, pp171—188. [9] J. Friedlander, H. Iwanies, What is the perity phenomenon?, http//www.ams.org/ notices/2009 07/ rtx090700817p.pdf [10] Jon Byl, On Resolving the Littlewood-Ross Paradox, Missouri Journal of Mathematical Sciences, Vol.12, No. 1, Winter 2000, pp.42–47 [11] J Harrison, formal proof --- theory and practice, Notices of AMS, 2008, http://www.ams.org/notices /200811/tx081101395p.pdf [12] J.R. Munkres, Topology, 2rd edition, Prentice Hall, Upper Saddle River, 2000. [13] Jon Pérez Laraudogoitia, "Supertasks", Stanford Encyclopedia of Philosophy, 2004, http://plato.stanford.edu/entries/spacetime supertasks/ [14] Jouko vaananrn, Second-order logic and foundations of mathematics, The bulletin of symbolic logic, Vol 7, 2001, pp.504—520. [15] K.Kuratowsky and A. Mostowsky, Set theory, With an introduction to descriptive set theory, North-Holland Publishingcom.1976, pp.118—120. [16] Fengsui Liu, On the prime k-tuple conjecture I, WSEAS Transactions on mathematics, Vol. 3, 2004, pp.744--747. [17] J. E. Littlewood, Littlewood's Miscellany (ed. Bela Bollobas) Cambridge University Press, Cambridge, 26, 1986 [18] Michiel Hazewinkel, Encyclopaedia of Mathematics,limit. http://eom.springer.de/l/l058820.htm

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