Generalizations of an Ancient Greek Inequality about the Sequence of Primes
aa r X i v : . [ m a t h . G M ] S e p Generalizations of an Ancient GreekInequality about the Sequence of Primes
Shaohua Zhang , Abstract
In this note, we generalize an ancient Greek inequality about thesequence of primes to the cases of arithmetic progressions even multi-variable polynomials with integral coefficients. We also refine Bouni-akowsky’s conjecture [16] and Conjecture 2 in [22]. Moreover, we givetwo remarks on conjectures in [22].
Keywords: inequality, primes, Euclid’s second theorem, Dirichlet’stheorem, Bouniakowsky’s conjecture
In his
Elements , Euclid proved that prime numbers are more than any as-signed multitude of prime numbers. In other words, there are infinitely manyprimes. For the details of proof, see [1, Proposition 20, Book 9]. Hardy andWright [2] called this classical result Euclid’s second theorem. Hardy likesparticularly Euclid’s proof. He [3] called it is ”as fresh and significant aswhen it was discovered—two thousand years have not written a wrinkle onit”. According to Hardy [3], ”Euclid’s theorem which states that the numberof primes is infinite is vital for the whole structure of arithmetic. The primesare the raw material out of which we have to build arithmetic, and Euclid’stheorem assures us that we have plenty of material for the task”. Andr´eWeil [4] also called ”the proof for the existence of infinitely many primesrepresents undoubtedly a major advance......”. Many people like Euclid’s1econd theorem. In his magnum opus
History of the Theory of Numbers ,Dickson [5] gave the historical list of proofs of Euclid’s second theorem fromEuclid (300 B.C.) to M´etrod (1917). Ribenboim [6] cited nine and a halfproofs of Euclid’s second theorem. The author [7] cited fifteen new proofs.Based on Euclid’s idea, people in Ancient Greek could prove that for n > Q i = ni =1 p i > p n +1 since p n +1 ≤ Q i = ni =1 p i −
1, where p i represents the i th prime.We call the inequality Q i = ni =1 p i > p n +1 Ancient Greek inequality. In 1907,Bonse [8] refined this inequality and proved that for n ≥ Q i = ni =1 p i > p n +1 and for n ≥ Q i = ni =1 p i > p n +1 . This kind of inequalities has been improvedsince then [9, 10]. Why are people interested in the inequality between Q i = ni =1 p i and p n +1 ? The main reason is of that this kind of inequalities areclosely related to the famous Chebychev’s function θ ( x ) = P p ≤ x log p . And θ ( x ) ∼ x ⇐⇒ π ( x ) ∼ x log x (The Prime Number Theorem).In a somewhat different direction, the aim of this note is to generalizethe ancient Greek inequality to the cases of arithmetic progressions evenmultivariable polynomials with integral coefficients. We noticed that p i canbe viewed as the i th prime value of polynomial f ( x ) = x . Let a and b be integers with a = 0, b > a, b ) = 1. Dirichlet’s classical andmost important theorem states that f ( x ) = a + bx can represent infinitelymany primes. Denote the i th prime of the form f ( x ) by P f,i . Naturally, wewant to prove that for every sufficiently large integer n , Q i = ni =1 P f,i > P f,n +1 ,where f = a + bx with a = 0, b > a, b ) = 1. More generally, wehope that if f is a multivariable polynomial with integral coefficients and f can take infinitely many prime values, then there is a constant C such thatwhen n > C , Q i = ni =1 P f,i > P f,n +1 . Thus, one could refine Bouniakowsky’sconjecture and so on. For the details, see Section 2. In this note, we always restrict that a k -variables polynomial with integralcoefficients is a map from N k to Z , where k ∈ N and N is the set of allpositive integers, Z is the set of all integers.Now, let’s begin with Bertrand’s and related problems in arithmeticprogressions. In 1845, Bertrand [5] verified for numbers < n > n − n . In1850, Chebychev [5] proved that there exists a prime between x and 2 x − x >
3. In the case of arithmetic progressions, Breusch [11], Ricci [12]2nd Erd¨os [13] proved respectively that for n ≥
6, positive integer, there isalways a prime p of the form 6 n + 1, and one of the form 6 n −
1, such that n < p < n . This implies immediately that the following Theorem 1 andTheorem 2. Theorem 1:
Let f ( x ) = 6 x + 1. Then P f, = 7, P f, = 13, P f, = 19,....And for n > Q i = ni =1 P f,i > P f,n +1 . Theorem 2:
Let f ( x ) = 6 x −
1. Then P f, = 5, P f, = 11, P f, = 17,....And for n > Q i = ni =1 P f,i > P f,n +1 .In 1941, Molsen [14] proved (1) for n ≥ n < p ≤ n always contains a prime of each of the forms 3 x + 1, 3 x −
1; (2) for n ≥ n < p ≤ n always contains a prime of each of the forms12 x + 1 , x − , x + 5 , x −
5. Based on Molsen’s work, it is not difficultto prove that the following theorems.
Theorem 3:
Let f ( x ) = 3 x + 1. Then P f, = 7, P f, = 13, P f, = 19,....And for n > Q i = ni =1 P f,i > P f,n +1 . Theorem 4:
Let f ( x ) = 3 x −
1. Then P f, = 2, P f, = 5, P f, = 11,....And for n > Q i = ni =1 P f,i > P f,n +1 . Theorem 5:
Let f ( x ) = 4 x + 1. Then P f, = 5, P f, = 13, P f, = 17,....And for n > Q i = ni =1 P f,i > P f,n +1 . Theorem 6:
Let f ( x ) = 4 x −
1. Then P f, = 3, P f, = 7, P f, = 11,....And for n > Q i = ni =1 P f,i > P f,n +1 .Let a and b be integers with a = 0, b > a, b ) = 1. In 1896,Ch. de la Vall´ee-Poussin [15] proved that P p ≡ a ( mod b ) ,p ≤ x log p equals xϕ ( b ) asymptotically. Therefore, for every sufficiently large integer n , P i = n +1 i =1 log P f,i equals P f,n +1 ϕ ( b ) asymptotically. Clearly, P f,n +1 ϕ ( b ) > P f,n +1 . It shows imme-diately that the following Theorem 7 holds. Theorem 7:
Let a and b be integers with a = 0, b > a, b ) = 1.And let f ( x ) = a + bx . Then there is a constant C depending on a and b such that when n > C , Q i = ni =1 P f,i > P f,n +1 .Based on the aforementioned theorems, also based on Bateman-Horn’sheuristic asymptotic formula [17], we give a strengthened form of Bouni-akowsky’s conjecture [16] which can be viewed as a refinement of special3orm of Schinzel-Sierpinski’s Conjecture [18] as follows: Conjecture 1: If f ( x ) is an irreducible polynomial with integral coef-ficients, positive leading coefficient, and there does not exist any integer n > f ( k ) for every integer k , then f ( x ) representsprimes for infinitely many x , moreover, there is a constant C such that when n > C , Q i = ni =1 P f,i > P f,n +1 .Conjecture 1 can be deduced by Bateman-Horn’s formula. Next, we willtry to generalize Conjecture 1 to the cases of multivariable polynomials withintegral coefficients. Firstly, we have the following theorems: Theorem 8 [19]:
Let f ( x, y ) = x + y + 1. Then there is a constant C such that when n > C , Q i = ni =1 P f,i > P f,n +1 . Theorem 9 [20]:
Let f ( x, y ) = x + y . Then there is a constant C suchthat when n > C , Q i = ni =1 P f,i > P f,n +1 . Theorem 10 [21]:
Let f ( x, y ) = x + 2 y . Then there is a constant C such that when n > C , Q i = ni =1 P f,i > P f,n +1 .By the aforementioned idea and theorems, one could strengthen a specialform of Conjecture 2 in [22] as follows: Conjecture 2:
Let f ( x , ..., x k ) be a multivariable polynomial with integralcoefficients, if there is a positive integer c such that for every positive integer m ≥ c , there exists an integral point ( y , ..., y k ) such that f ( y , ..., y k ) > Z ∗ m = { x ∈ N | gcd( x, m ) = 1 , x ≤ m } , and there exists an integral point( z , ..., z k ) such that f ( z , ..., z k ) ≥ c is prime, then f ( x , ..., x k ) representsprimes for infinitely many integral points ( x , ..., x k ). Moreover, there is aconstant C such that when n > C , Q i = ni =1 P f,i > P f,n +1 . Remark 1:
Conjecture 2 implies that a special case of Conjecture 1 in [22].Namely, if f ( x , ..., x k ) is a multivariable polynomial with integral coeffi-cients, and represents primes for infinitely many integral points ( x , ..., x k ),then there is always a constant c such that for every positive integer m > c ,there exists an integral point ( y , ..., y k ) such that f ( y , ..., y k ) > Z ∗ m .In fact, by Conjecture 2, we know that there is a constant C such that when n > C , Q i = ni =1 P f,i > P f,n +1 . Let C < k ≤ C + 1 and let c = P f,k . When m > c , we can assume that c ≤ P f,k + h ≤ m < P f,k + h +1 with h ≥
0. If forsome 1 ≤ r ≤ k + h , gcd( P f,r , m ) = 1, then there exists an integral point( y , ..., y k ) such that f ( y , ..., y k ) = P f,r is in Z ∗ m . If for any 1 ≤ r ≤ k + h ,4cd( P f,r , m ) >
1, then m ≥ Q i = k + hi =1 P f,i > P f,k + h +1 since C < k ≤ k + h and gcd( P f,i , P f,j ) = 1 for i = j . It is a contradiction. Remark 2:
Conjecture 1 can not be extended to arbitrary number-theoreticfunctions without a proviso. For example, let h ( n ) = p = 2 , n = 1 p = 3 , n = 2 ...... the least prime of the form k × Q i = n − i =1 p i + 1 , n ≥ . Clearly, for any positive integer n , Q i = ni =1 P h,i < P h,n +1 , where P h,i = h ( i )is the i th prime value of the function h ( n ).By this example, one also can find that Conjecture 1 in [22] can notbe extended to arbitrary number-theoretic functions without a proviso. Infact, if there is such a constant c , then there is always a positive integer k such that c < h ( k ). Let m = Q i = ki =1 P h,i . Clearly, in this case, c < m ≤ h ( k + 1) − < h ( k + 1) and there does not exist any positive integer y such that h ( y ) > Z ∗ m . Otherwise, y ≥ k + 1. It is impossible since m < h ( k + 1).Let s ≥ k ≥ f ( x , ..., x k ) , ..., f s ( x , ..., x k )be multivariable polynomials with integral coefficients. We also assumethat f ( x , ..., x k ) , ..., f s ( x , ..., x k ) represent simultaneously primes for in-finitely many integral points ( x , ..., x k ). Denote the set of integral points( x , ..., x k ) such that f ( x , ..., x k ) , ..., f s ( x , ..., x k ) are primes by X . Let β f, = Q i = si =1 f i ( X ), where X ∈ X such that the norm || ( f ( X ) , ..., f s ( X )) || is the least. Let β f, = Q i = si =1 f i ( X ), where X ∈ X such that gcd( β f, , β f, ) =1, || ( f ( X ) , ..., f s ( X )) || < || ( f ( X ) , ..., f s ( X )) || ≤ || ( f ( X ) , ..., f s ( X )) || with X ∈ X, X = X , X = X and gcd( Q i = si =1 f i ( X ) , β f, ) = 1, ... Let β f,j = Q i = si =1 f i ( X j ), where X j ∈ X such that for any 1 ≤ r ≤ j − β f,j , Q i = si =1 f i ( X ) × ... × Q i = si =1 f i ( X j − )) = 1, || ( f ( X r ) , ..., f s ( X r )) || < || ( f ( X j ) , ..., f s ( X j )) || ≤ || ( f ( X ) , ..., f s ( X )) || with X ∈ X, X = X , X = X , ..., X = X j and gcd( Q i = si =1 f i ( X ) , Q i = si =1 f i ( X ) × ... × Q i = si =1 f i ( X j − )) = 1,... Clearly, gcd( β f,i , β f,j ) = 1 for i = j . Notice that pairwise distinct primesare pairwise relatively prime. The sequence of primes { p i } has a beautifulproperty: if any integral sequence 1 < a < ...a n < ... with gcd( a i , a j ) = 1for i = j , then p i ≤ a i for any positive integer i . For the proof of this prop-erty, see Appendix. Therefore, like the i th prime p i , β f,i can be viewed as5he i th ”desired prime number”. Thus, one could give a strengthened formof Conjecture 2 in [22] as follows: Conjecture 3:
Let f ( x , ..., x k ), ..., f s ( x , ..., x k ) be multivariable poly-nomials with integral coefficients, if there is a positive integer c such thatfor every positive integer m ≥ c , there exists an integral point ( y , ..., y k )such that f ( y , ..., y k ) > , ..., f s ( y , ..., y k ) > Z ∗ m = { x ∈ N | gcd( x, m ) = 1 , x ≤ m } , and there exists an integral point ( z , ..., z k ) suchthat f ( z , ..., z k ) ≥ c, ..., f s ( z , ..., z k ) ≥ c are all primes, then f ( x , ..., x k ),..., f s ( x , ..., x k ) represent simultaneously primes for infinitely many integralpoints ( x , ..., x k ). Moreover, there is a constant C such that when n > C , Q i = ni =1 β f,i > β f,n +1 . In this note, we generalized an ancient Greek inequality about the sequenceof primes to the cases of arithmetic progressions. By Bateman-Horn’s heuris-tic asymptotic formula and also based on the work of Motohashi Yoichi,Friedlander John, Iwaniec Henryk, Heath-Brown, and so on, we refinedBouniakowsky’s conjecture and Conjecture 2 in [22]. Knuth called Euclid’sAlgorithm the granddaddy of all algorithms. Based on the work in this note,one can see that the Ancient Greek inequality about the sequence of primesalso is the granddaddy of the inequalities about the sequence of some kindspecial kinds of primes.
Thank my advisor Professor Xiaoyun Wang for her valuable help. Thankfor the Institute Advanced Study in Tsinghua University for providing mewith excellent conditions. This work was partially supported by the Na-tional Basic Research Program (973) of China (No. 2007CB807902) and theNatural Science Foundation of Shandong Province (No. Y2008G23). [1] Thomas Little Heath, The Thirteen Books of the Elements, translatedfrom the text of Heiberg with introduction and commentary, CambridgeUniv. Press, Cambridge (1926). 62] Hardy G. H. and Wright E. M., An Introduction to the Theory of Num-bers, Oxford: The Clarendon Press, (1938).[3] G. H. Hardy, A Mathematician’s Apology, Cambridge University Press,(1940).[4] Andr´e Weil, Number theory, an approach through history from Hammu-rapi to Legendre. Birkh¨aser, Boston, Inc., Cambridge, Mass., (1984).[5] Leonard Eugene Dickson, History of the Theory of Numbers, Volume I:Divisibility and Primality, Chelsea Publishing Company, New York, (1952).[6] Paulo Ribenboim, The book of prime number records, Springer-Verlag,New York, (1988).[7] Shaohua Zhang, Euclid’s Number-Theoretical Work, available at:http://arxiv.org/abs/0902.2465[8] H. Bonse, ¨Uer eine bekannte Eigenschaft der Zahl 30 und ihre Verallge-meinerung, Arch. Math. Phys., 12, 292-295, (1907).[9] P´osa Lajos, ¨Uber eine Eigenschaft der Primzahlen, (Hungarian) Mat.Lapok, 11, 124-129, (1960).[10] Panaitopol Laurent¸iu, An inequality involving prime numbers, Univ.Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 11, 33-35, (2000).[11] R. Breusch, Zur Verallgemeinerung der Bertrandschen Postulates dasszwischen x und 2x stets Primzahlen liegen, Math. Z., 34, 505-526, (1932).[12] G. Ricci, Sul teorema di Dirichlet relativo alla progresione aritmetica,Boll. Un. Mat. Ital., 12, 304-309, (1933).[13] P. Erd¨os, ¨Uber die Primzahlen gewisser arithmetischen Reihen, Math.Zeit., 39, 473-491, (1935).[14] K. Molsen, Zur Verallgemeinerung der Bertrandschen Postulates. DeutscheMath. 6, 248-256, (1941).[15] Ch. de la Vall´ee-Poussin, Recherches analytiques sur la th´eorie desnombres (3 parts), Ann. Sec. Sci. Bruxelles, 20, 183-256; 361-397, (1896).[16] Bouniakowsky, V., Nouveaux th´eor`emes relatifs `a la distinction desnombres premiers et `ala d´e composition des entiers en facteurs, Sc. Math.Phys., 6, 305-329, (1857). 717] Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formulaconcerning the distribution of prime numbers, Math. Comp., Vol. 16, No.79,363-367, (1962).[18] A. Schinzel and W. Sierpinski, Sur certaines hypotheses concernant lesnombres premiers, Acta Arith., 4 (1958), 185-208, Erratum 5, 259, (1958).[19] Motohashi Yoichi, On the distribution of prime numbers which are ofthe form x + y + 1, Acta Arith., 16, 351-363, (1969/1970).[20] Friedlander John, Iwaniec Henryk, The polynomial x + y captures itsprimes, Ann. of Math., (2) 148, no. 3, 945-1040, (1998).[21] Heath-Brown D. R. Primes represented by x + 2 y , Acta Math., 186,No. 1, 1-84, (2001).[22] Shaohua Zhang, On the Infinitude of Some Special Kinds of Primes,available at: http://arxiv.org/abs/0905.1655 In this appendix, we prove the following theorem 11:
Lemma 1: π ( n ) is the largest among the cardinality of all sub-sets in whicheach element exceeds 1 and pairwise distinct elements are pairwise relativelyprime of { , , ..., n } , where π ( x ) represents the number of primes less thanor equal to x . Proof:
Easy. Let S be a sub-set of { , , ..., n } such that in S , each ele-ment exceeds 1 and pairwise distinct elements are pairwise relatively prime.Denote the cardinality of S by | S | . If | S | > π ( n ), then Q x ∈ S x has at least π ( n ) + 1 distinct prime divisors. This implies that there must be an element a ∈ S such that a ≥ p π ( n )+1 > n . It is a contradiction since S ⊆ { , , ..., n } .This complets the proof of Lemma 1. Theorem 11:
If any integral sequence 1 < a < ...a n < ... with gcd( a r , a j ) =1 for r = j , then p i ≤ a i for any positive integer i , where p i is the i th prime. Proof:
Easy. For any positive integer i , we consider the set S = { a , ..., a i } .By known condition, we have gcd( a r , a j ) = 1 for r = j . Namely, in S ,each element exceeds 1 and pairwise distinct elements are pairwise relativelyprime. So, by Lemma 1, i ≤ π ( a i ). It shows that p ii
Easy. For any positive integer i , we consider the set S = { a , ..., a i } .By known condition, we have gcd( a r , a j ) = 1 for r = j . Namely, in S ,each element exceeds 1 and pairwise distinct elements are pairwise relativelyprime. So, by Lemma 1, i ≤ π ( a i ). It shows that p ii ≤ a ii