Note On The Maximal Prime GapsN. A. Carella

Abstract: This note presents a result on the prime gap of the form , where is a constant, for any arbitrarily small real number , and all sufficiently large integer

. Equivalently, the result shows that short intervals contain prime numbers for

all sufficiently large real numbers , and unconditionally. An application

demonstrates that a prime can be determined in deterministic polynomial time .

Mathematics Subject Classification: 11A41, 11Y55, 11N05, 11N13, 11P32.Keywords: Prime Number, Prime Gap, Short Interval, Prime Complexity, Primality Test, Deterministic Polynomial Time.

1. Introduction Let be an integer, and let denotes the nth prime number. The nth prime gap is defined as the

difference of two consecutive prime numbers. This note investigates the maximal size of the prime gap function. For a large real number , the maximal prime gap function is defined by

(1) .

The average prime gap of a short sequence of consecutive primes is given by the asymptotic formula

(2)

This immediately shows that . Some of the leading prime gaps open problems provide upper bounds and lower bounds of the maximal prime gap.

Cramer Prime Gap Problem. This problem claims that

(3)

for some constant , see [CR]. The precise implied constant in this formula has been debated in

the literature, this conjecture is discussed in [GL].

Erdos-Rankin Prime Gap Problem. This problem claims that

(4)

for some constant , and that the constant cannot be replaced with any increasing function of , see [ER], [ES], [RA], [MP]. The notation denotes the k-fold

iterations of the logarithm.

The Cramer conjecture provides an upper bound of the maximal prime gap function . In comparison, the Erdos conjecture provides a lower bound of the maximal

size of the prime gap function . Extensive details on the theory of maximal prime gap and primes in short intervals are given in [RN], [BK], [HL], [SD], [SG], [GH], [GL], [GK], [OK], et alii.

2. The Proof of the ResultThe numerical evidence, [NT], [PR], and the result within suggest that the Erdos conjecture is closer to the correct order of magnitude of the maximal prime gap function

(5)

as seemed to be confirmed below.

Theorem 1. Let be an arbitrarily small real number, and let be an integer. Then, the nth prime gap satisfies

(6) ,

where is a constant, for all sufficiently large integer .

Proof: Given a small real number , let be an integer, and let . By Lemma 4, in Section 3, the inequality

(7)

is satisfied for all sufficiently large integer . Moreover, taking logarithm of both sides, and simplifying yield

(8) ,

where is a constant. Apply the asymptotic formula for the nth prime , see (21), [DU], [NW, p. 254], and simplify to obtain

(9) ,

where

The result in Theorem 1 is equivalent to the existence of prime numbers in very short intervals for all sufficiently large real numbers , unconditionally.

This assertion follows from the inequality

(10) ,

for . Prime pairs with large prime gaps near are studied in [MW], and large prime gaps near in arithmeritc progressions

are studied in [GM].

2.1 Density of Primes in Short IntervalsThe density of primes on large intervals , where , has the expected order of magnitude

(11) , see [DF, p. 70], and [HB]. In contrast, the precise form of the counting function for primes in short intervals with remains unknown. In fact, the density of prime numbers in very short intervals does not satisfy (11) since it has large flutuations. More precisely, for any , the inequalities

(12)

occur infinitely often as

Corollary 2. Let be an arbitrarily small real number, let be an integer, and let be a sufficiently large real number. Then, there are at least

(13)

prime numbers in the very short interval .

A related problem is the solution of the correct inequality

(14)

for as . This is an important problem in the theory of primes numbers, known as theHardy-Littlewood conjecture. It asks whether or not the intervals of lenghts containmore primes than the intervals as tends to infinity, see [ER, p. 6], [GD, p. 24], [HR]. Accordding to a discussion in [DF, p. 217], the numerical calculations for a potential counterexample is beyond the reach of current machine computations. For small multiple , there isan equality .

2.2 Merit Factor The merit factor of a pair of consecutive primes is defined by the ratio . The studyof the merit factor has a prominent place in prime number theory, [RN, p. 254]. The average value in (2) immediately implies that . The limit supremum wass determined about 80 years ago by Westzynthius, and the limit infimum is a recent development, refer to [GK], [PT] for full detail. These limits are expressed in the relations

(15) and

respectively. Some historical information on the calculations of these limits appears in [NW, p. 340]. Aconsequence of Theorem 1 is an upper bound of the merit factor, that is,

(16) .

2.3 Maier-Pomerance Integer Gap Problem Let be the maximal of the Jacobsthal function. This problem claims that

(17) ,

confer [MP, p. 202] and [HN]. The relevance of this problem arises from the inequality . Since for any constant , the result in

Theorem 1 is in line with this conjecture. Furthermore, the proof of Lemma 4, and Erdos conjecture (4) suggest that the maximal prime gaps should be something of the form

for some constant . In synopsis, the inequality

(18)

is plausible. But this could be far more difficult to prove. A graph of the function

(19)

for , supplied on the last page, demonstrates that it is nonnegative , and increasing for , it also demonstrates that is significantly larger than the nth prime gap as .

2.4 Least Quadratic Nonresidue ProblemLet , and let be a character modulo , the least quadratic nonresidue is defined by

. Some partial new results for the least quadratic nonresidue for some

characters are demostrated in [BL, Corollary 2] , for example, . This result is strikingly similar to the upper bound stated in Theorem 1.

3. Supporting MaterialsThe basic background information and a few Lemmas are provided in this section. The proofs of these Lemmas use elementary methods. At most a weak form of the Prime Number Theorem

(20)

is required, see [RN]. The innovation here is splicing together these elementary concepts into a coherent and effective result in the theory of primes numbers in very short intervals.

3.1 Formula for the nth Prime, And Associated SeriesThe Prime Number Theorem (20) implies that the nth prime has an asymptotic expression of the form

, [RN, p. 249], [HW, p. 459], [DU]. The Cipolla formula specifies the exact formula for the nth prime

(21) .

The function has the power series expansion

,

(22)

where is a polynomial, and is an

abbreviated notation for the sth power respectively. For example, , , and

. The leading coefficient of the sth polynomial has alternating sign for . Other details on the asymptotic representation of the nth prime are discussed in [AR, p. 27], [CL], and [DU].

The function is quite similar to the simpler function

(23)

,

and the asymptotic analysis of these two functions are similar. An important component in the proof of Lemma 3 is the behavior and magnitude of the logarithmic ratio as . The result below works out a lower and an upper estimate of the logarithm ratio of the function associated with the nth prime .

Lemma 3. Let be a sufficiently large integer. Then

(i) ,

(ii) ,

(iii) ,

where are constants.

Proof: (i) The derivative of the function satisfies on the interval for . Hence, the difference is nonnegative on the interval .

(ii) This follows from the power series (22).

(iii) Start with the ratio

(24) .

Applying the inequality , where , and , leads to

(25) ,

where is a constant. In addition, . Consequently

(26) ,

where are constants. The lower and upper bounds are computed in a term by term basis. Forexample, the difference of the first term of the power series (22) satisfies

(27) ,

where are constants. Lastly, as the function is an alternating power series, the first term is sufficient.

The upper bound of the logarithmic ratio is an effective way of estimating the magnitude of the error term in the difference . It sidesteps the difficulty encountered whileusing the asymptotic formula or similar asymptotic formula.

3.2 The Prime Numbers Equations And Inequalities The exponential prime numbers equations and inequalities

(28)

and other variations of these equations and inequalities have been studied by many workers in the field, confer [GD, p. 19], and [RN, p. 256] for discussions and references. The specific case

, which implies the Cramer conjecture, is easy to establish using the elementary method employed in this work. A stronger result, as established here, seems to be plausible.

Lemma 4. Let be an arbitrarily small real number, and let be a sufficiently large integer.Then

(29) .

Proof: Write a pair of consecutive primes as

(30) where is the power series (22), and consider the inequality

(31) ,

where is a function of . Replacing (30) into (31) transforms it into

.

(32)

Now use , where , and , to reduce it to

(33) ,

where are constants. In Lemma 3 it is shown that

(34) and ,

where are constants. Accordingly, the penultimate inequality can be written as

(35) .

The term on the right side of the previous inequality dominates if the parameter isset to for any arbitrarily small real number . Therefore, the inequality (31) is satisfied by any sufficiently large integer

4. ApplicationsThere are many practical and theoretical applications of Theorem 1 in the Mathematics, Cryptography, and Physics. Two obvious applications are considered here.

4.1 Generating Prime NumbersRecent work on the complexity of generating a prime number appears in [TH], [HW, p. 13].

Corollary 5. Given any real number , a prime can be constructed in deterministic polynomial time complexity of arithmetic operations. Proof: Apply the AKS primality test algorithm to the sequence of consecutive odd numbers

,

where , and . By Theorem 1, at least one of these integers is a prime number. Thus, repeating the primality test times, result in the running time complexity of

The AKS primality test algorithm appears in [AK], it has deterministic polynomial time complexity of arithmetic operations. An algorithm of faster running time appears in [LT].

4.2 Scherk Expansions of PrimesThe binary expansion of prime number is computable in polynomial time complexity, but the Scherk expansion of a prime has exponential time complexity, and it seems to be impossible to reduce it to a lower time complexity. This is an expansion of the form, [NW, p. 353],

(36) or ,

where , and . A proof of this expansion is given in [BN]. For example, the Scherk expansion of the prime number

requires about primes to complete. This amounts to a very large scale computation. However, the short Scherk expansion

has polynomial time complexity, this follows from Corollary 5.

5. Numerical DataSignificant amount of data have been complied by several authors over the past decades, see [NT], [NL],[PR], [TS], et alii, for full details.

A few examples of prime gaps, and the estimated upper bounds are demonstrated here. These estimates utilize the inequality

(37) ,

with , see Theorem 1, and the Maier-Pomerance problem in Section 1 for explanations.

(a) The largest prime gap for the prime listed on the tables in [NL] satisfies the inequality:

(b) The recently reported very large prime gap of 337446 for the 7976 digits prime , see [WK], satisfies the inequality:

.

(c) A few others reported examples for the prime gaps of very large probable primes also satisfy the inequality, in particular,

,

for a 86853 decimal digits probable prime , see [PR], [WK].

A graph of the function

(38)

for is supplied to demonstrate that it is nonnegative , and increasing for , it also demonstrates that is significantly larger than the nth prime gap

as .

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[ES] Erds, P. The difference of consecutive primes. Quart. J. Math. Oxford Series 6, 1935, 124-128.[ER] Erds, P. Problems and results in number theory. Recent progress in analytic number theory, Vol.

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[HL] Huxley, M. N. On the difference between consecutive primes. Invent. Math. 15 (1972), 164-170.[HN] Hagedorn, Thomas R. Computation of Jacobsthal's function h(n) for n

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