Gap Between Consecutive Primes By Using A New Approach

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1 Gap Between Consecutive Primes By Using A New Approach Hassan Kamal hassanamal.edu@gmail.com Abstract It is nown from Bertrand s postulate, whose proof was first given by Chebyshev, that the interval [x, 2x] contains at least one prime number, but the best estimate of the length of an interval [x, cx] that must contain a prime number, is certainly given by values of c significantly smaller. Baer, Harman, and Pintz, proved that c x for all large x. Dusart proved for x that c This paper gives a 25 proof for c exp ϑx x, for all large x, where ϑ is the first Chebyshev function. 1 Introduction The prime number theorem, proved in 1896, states that the gap between a prime p and the next prime has the size about p on average. The effective length of the gap might be much more or less than this. However, the prime number theorem allows to deduce an upper bound on the length of an interval [x, cx] that must contain a prime number: for every c > 1, there is a number x 0 such that [x, cx] contains a prime for all x > x 0. In Hadamard s An essay on the psychology of invention in the mathematical field, we can appreciate: It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one. However, an elementary approach might also be efficient, it depends on the form of its formulation. We use elementary methods to obtain the proof of the existence of prime numbers in all intervals [x, cx] for all large x with c very close to 1, i.e., c exp ϑx x. Roughly speaing, the length of the interval [x, cx], scarcely exceeds the maximal prime gaps. All maximal gaps between primes are now nown, up to low 19-digit primes A This data apparently supports our results. Let denote the th prime number with 1, and let ϕ be the function defined as follows: ϕ x : x ϑ x x for all real x > 2, where ϑ is the first Chebyshev function defined as well nown by ϑ x x p p2 1

2 where p x runs over primes. Then one can write ϕ ϑ 1, for all > 1. Throughout the article, it is worth mentioning, that it is useful to show the monotonicity of the quantity for 1, where we define as follows: +1 : ϑ 1 With the notation used in this paper, we assume that it is strongly convenient to rewrite a connection between two consecutive prime numbers, in a form such that +1 c. Whereas if > 1, it is nown that , consequently c 2, moreover, we can observe c c if and only if. 2 The results Lemma 1. The double inequality is true for all Proof. We begin showing the first inequality, since +1 ϑ c c c and we now that c 2, we get clearly +1 + c 2

3 By using the nown Bonse s inequality [2], in logarithmic form, we obtain directly for all > 4, Therefore, we deduce c and we get clearly for all > 4. We now focus on the second inequality of Lemma 1, since +1 c ϑ 1 ϑ 1 c It now suffices for our purpose to prove the condition: c ϑ 1 c 1 ϑ 1 The fact that x 1 x 1 X 1 X if 0 x 1 X, enables us to obtain c c By using the above inequality with the needed steps, we prove exactly the condition satisfying the second inequality of Lemma 1. We firstly deduce 1 c 3 c 2

4 And by adding 1 to both sides of the inequality 2, we obtain the condition previously T required, which allows us to have +1 1, for all > 4. In addition to this, we verify computationally Lemma 1 even for values of less or equal than 4, and this completes the proof. Lemma 2. The inequality holds for all large Proof. Let introduce a suitable substitution by putting e λ, where λ is the real such that ϑ λ. As we see, λ depends strictly on the distribution of prime numbers. Dusart [3] established an upper bound on ϑx. The author obtained λ for x any 1. Then we get the expression +1 e λ c eλ c p 2 e λ p 2 eλ c p In order to mae an upper bound on +1 +1, we assume a result due to Baer, Harman, and Pintz [1], about an upper bound on c, and we can use appropriately the double inequality p n c n p n for all sufficiently large n. It follows that e λ c eλ p 2 e λ eλ c e λ 2 e λ Let f be the function defined as follows: fx e Λx x 2 x e Λx x x 1 + c eλ p 2 e λ p eλx x 2 e Λx x x p x x

5 For any given Λ such that 0.99 Λ , we can assume that Λ λ is always greater than 0.99 for sufficiently large x. With enough effort, we can verify that fx is increasing and tends to 1 as x goes to infinity, in other words, we deduce fx 1, which implies that Theorem 3. For all sufficiently large x, there exists at least one prime number between x and x exp. ϑx x Proof. In Introduction 1, as by definition, for all > 1, we have ϕ ϑ 1 + Therefore, ϕ ϕ c +1 + p We now apply the previous lemmas, and we deduce clearly, for sufficiently large that ϕ ϕ +1 > 0. Since ϕ c 1 + ϑ 1 ϕ It follows Equivalently c ϑ 1 2 p c exp ϑ 1 5

6 Hence 2 p +1 exp p 1+ϕ ϑ 1 However, ϕ ϕx for x +1, this fact implies the following inequalities: Since We get more precisely x +1 p 1+ϕ We conclude that between x and x exp x 1+ϕx x 1+ϕx x exp ϑx x x +1 x exp ϑx x ϑx x, exists at least the prime +1. Corollary 4. For all sufficiently large x, there exists at least one prime number between x and x exp x x Proof. According to Theorem 3, for all sufficiently large x, there exists a prime number p such that x p x exp. ϑx x For all x , Rosser and Schoenfeld [4] provided explicitly a lower bound on ϑx, that is, 0.998x ϑx, since x exp ϑx x for all x We finally deduce the explicit result: for all sufficiently large x. x exp 0.998x x x p x exp 0.998x x 3 Better results If we use a sharper lower bound on ϑx for large x, it will be possible to provide better results concerning the prime gaps. In fact, Theorem 3 suggests, that every improvement about the length of an interval [x, cx] that must contain a prime, depends strictly on a sharper lower bound on ϑx. 6

7 4 Acnowledgments The author is grateful to the referee for valuable comments and interesting suggestions. References [1] R. C. Baer, G. Harman, and J. Pintz, The difference between consecutive primes, Proc. London Math. Soc , [2] H. Bonse, Über eine beannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung, Arch. Math. Phys , [3] Pierre Dusart, Estimates of some functions over primes without R.H, preprint, 2010, [4] J. Barley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θx and ψx, Math. Comp., , Mathematics Subject Classification: Primary 11A41; Secondary 11N05. Keywords: prime number, distribution of primes. Concerned with sequence A