In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]
Deep Dive into Generalizations of an Ancient Greek Inequality about the Sequence of Primes.
In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky’s conjecture [16] and Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]
In his Elements, Euclid proved that prime numbers are more than any assigned multitude of prime numbers. In other words, there are infinitely many primes. For the details of proof, see [1,Proposition 20,Book 9]. Hardy and Wright [2] called this classical result Euclid's second theorem. Hardy likes particularly Euclid's proof. He [3] called it is "as fresh and significant as when it was discovered-two thousand years have not written a wrinkle on it". According to Hardy [3], "Euclid's theorem which states that the number of primes is infinite is vital for the whole structure of arithmetic. The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task". André Weil [4] also called "the proof for the existence of infinitely many primes represents undoubtedly a major advance......". Many people like Euclid's second theorem. In his magnum opus History of the Theory of Numbers, Dickson [5] gave the historical list of proofs of Euclid's second theorem from Euclid (300 B.C.) to Métrod (1917). Ribenboim [6] cited nine and a half proofs 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 > 1, i=n i=1 p i > p n+1 since p n+1 ≤ i=n i=1 p i -1, where p i represents the i th prime. We call the inequality i=n i=1 p i > p n+1 Ancient Greek inequality. In 1907, Bonse [8] refined this inequality and proved that for n ≥ 4, i=n i=1 p i > p 2 n+1 and for n ≥ 5, i=n i=1 p i > p 3 n+1 . This kind of inequalities has been improved since then [9,10]. Why are people interested in the inequality between i=n i=1 p i and p n+1 ? The main reason is of that this kind of inequalities are closely related to the famous Chebychev’s function θ(x) = 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 generalize the ancient Greek inequality to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We noticed that p i can be viewed as the i th prime value of polynomial f (x) = x. Let a and b be integers with a = 0, b > 0 and gcd(a, b) = 1. Dirichlet’s classical and most important theorem states that f (x) = a + bx can represent infinitely many primes. Denote the i th prime of the form f (x) by P f,i . Naturally, we want to prove that for every sufficiently large integer n, i=n i=1 P f,i > P f,n+1 , where f = a + bx with a = 0, b > 0 and gcd(a, b) = 1. More generally, we hope 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 that when n > C, i=n i=1 P f,i > P f,n+1 . Thus, one could refine Bouniakowsky’s conjecture and so on. For the details, see Section 2.
In this note, we always restrict that a k-variables polynomial with integral coefficients is a map from N k to Z, where k ∈ N and N is the set of all positive integers, Z is the set of all integers. Now, let’s begin with Bertrand’s and related problems in arithmetic progressions. In 1845, Bertrand [5] verified for numbers < 6000000 that for any integer n > 6 there exists at least one prime between n -2 and n 2 . In 1850, Chebychev [5] proved that there exists a prime between x and 2x -2 for x > 3. In the case of arithmetic progressions, Breusch [11], Ricci [12] and Erdös [13] proved respectively that for n ≥ 6, positive integer, there is always a prime p of the form 6n + 1, and one of the form 6n -1, such that n < p < 2n. This implies immediately that the following Theorem 1 and Theorem 2. Molsen [14] proved (1) for n ≥ 199, the interval n < p ≤ 8 7 n always contains a prime of each of the forms 3x + 1, 3x -1; (2) for n ≥ 118, the interval n < p ≤ 4 3 n always contains a prime of each of the forms 12x + 1, 12x -1, 12x + 5, 12x -5. Based on Molsen’s work, it is not difficult to prove that the following theorems. And let f (x) = a + bx. Then there is a constant C depending on a and b such that when n > C, i=n i=1 P f,i > P f,n+1 . Based on the aforementioned theorems, also based on Bateman-Horn’s heuristic asymptotic formula [17], we give a strengthened form of Bouniakowsky’s conjecture [16] which can be viewed as a refinement of special form of Schinzel-Sierpinski’s Conjecture [18] as follows:
Conjecture 1: If f (x) is an irreducible polynomial with integral coefficients, positive leading coefficient, and there does not exist any integer n > 1 dividing all the values f (k) for every integer k, then f (x) represents primes for infinitely many x, moreover, there is a constant C such that when n > C, i=n i=1 P f,i > P f,n+1 . Conjecture 1 can be deduced by Bateman-Horn’s formula. Next, we will try to generalize Conjecture 1 to the cases of multivariable polynomials with integral coefficients. Firstly, we have the following theorems:
Theorem 8 [19]: Let f (x, y) = x 2 + y 2 + 1. Then there is a constant C such
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