THE MAXIMAL PRIME GAPS SUPREMUM AND THE FIROOZBAKHT’S HYPOTHESIS N 30

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a research in the subject of the maximal prime gaps. The document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht’s Hypothesis N 30. Firoozbakht Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously the Firoozbakht’s Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht. c ⃝2014 Jan Feliksiak 2000 Mathematics Subject Classification. 0102, 11A41, 11K65, 11L20, 11N05, 1102, 1103.


Preliminaries.
Within the scope of the paper, prime gap of the size g ∈ N | g ≥ 2 is defined as an interval between two primes (p i , p i+1 ], containing (g − 1) composite integers. Maximal prime gap of the size g, is a gap strictly exceeding in size any preceding gap. All calculations and graphing were carried out with the aid of the M athematica software. For all n ∈ N | n ≥ 8, we make the following definitions: The bounds on the logarithm of n! are given by: (1.1) n log (n) − n + 1 ≤ log (n!) ≤ (n + 1) log (n + 1) − n ∀n ∈ N | n ≥ 5 Proof. Evidently, Hence, the pertinent integrals to consider are: Accordingly, evaluating those integrals we obtain: Concluding the proof of Lemma 1.6.

Remark 1.2.
Observe that log M (t) is a difference of logarithms of factorial terms: log Consequently, implementing the lower/upper bounds on the logarithm of n! for the bounds on log M (t) , results in bounds of the form: (1.5) log Keeping the values of c, n and t constant and letting the variable k to increase unboundedly, results in an unboundedly decreasing function. When implementing the lower/upper bounds on the logarithm of n! for the Supremum/Infimum bounds on log M (t) , the variable k appears only with values k = {0, 1} respectively. The combined effect of the difference of the logarithms of factorial terms in log M (t) and the decreasing property of the function 1.5, imposes a reciprocal interchange of the bounds 1.1, when implementing them for the bounds on log M (t) .
The Supremum Bound on the logarithm of the binomial coefficient M (t) is given by: Proof. From Lemma1.6 we have: Substituting from the inequality 1.7 into the Definition 1.4 we obtain: Consequently, The Supremum bound UB (t) produces an increasing, strictly monotone sequence in R. At n = 8, the difference UB (t) − log M (t) attains 0.197362 and diverges as n → ∞. Therefore, Lemma 1.7 holds as specified.
Consequently, from Lemma 1.8 and 1.7 we have for all n ∈ N | n ≥ 8: (1.14) log

Maximal prime gaps
From the Prime Number Theorem we have that an average gap between consecutive primes is given by log n for any n ∈ N. There exist however prime gaps much shorter -containing only a single composite number, and gaps which are much longer than average -the maximal prime gaps.
We begin with a preliminary derivation. Since the integers from 1 to n contain ⌊ n p ⌋ multiples of the prime number p, ⌊ n p 2 ⌋ multiples of p 2 etc. Thus it follows that: In accordance with the definitions 1.1 of Gs (n) , 1.2 of t and 1.3 of M (t) we obtain: and so by the above, Lemma 1.7 and 1.8 we have: Where p is as usual a prime number. Let's define: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 June 2020 doi:10.20944/preprints202006.0366.v1 The case when there does not exist any prime factor p of M (t) within the interval from n to (n + Gs (n) ) = t for any n ∈ N | n ≥ 11, imposes an upper limit on all prime factors p of M (t) . Consequently in this particular case, every prime factor p must be less than or equal to s = ⌊ t 2 ⌋ .

Proof.
Let p be a prime factor of M (t) so that K p ≥ 1 and suppose that every prime factor p ≤ n. If s < p ≤ n then, p < (n + Gs (n) ) < 2p and and so K p = 0. Therefore p ≤ s for every prime factor p of M (t) , for any n ∈ N | n ≥ 11.

Maximal Prime Gaps Supremum.
The bounds on the logarithm of M (t) are given by Lemma 1.7 and 1.8: Remark 2.1.
• The proof of the Maximal Prime Gaps Supremum implements the Supremum bound function UB (ts) . Due to the fact that the function UB (t) applies values of n, c and t directly, it imposes a requirement to generate a set of pertinent values to correctly approximate the interval s. This is to ascertain that the generated interval is at least equal or greater than s as given by Definition 2.1, as well as the corresponding value of c. Respective definitions follow: For any n ∈ N | n ≥ 11 there exists at least one p ∈ N | n < p ≤ n+Gs (n) = t; where p is as usual a prime number and the maximal prime gaps standard measure Gs (n) is given by:

Proof.
Suppose that there is no prime within the interval from n to t. Then in accordance with the hypothesis, by Lemma 2.2 we have that every prime factor p of M (t) must be less than or equal to s = ⌊ t 2 ⌋ . Invoking Definitions 2.3, 2.4 and 2.5, Lemma 1.7, 1.8 and the inequality 2.1 we obtain in such a case, for all n ∈ N | n ≥ 11: In accordance with the hypothesis therefore, it must be true that: (2.5) log However, at n = 47 the difference 2.5 attains ∼ 7.69823 and diverges as n increases unboundedly. Since the difference generates a positive sequence in R, we apply therefore the Cauchy's Root Test for n ≥ 47: At n = 47 the Cauchy's Root Test attains ≈ 1.20947 and tends asymptotically to 1 decreasing strictly from above. Thus, by the definition of the Cauchy's Root Test, the series formed from the terms of the difference LB (t) − UB (ts) diverges to infinity as c increases unboundedly. This implies that for all n ∈ N | n ≥ 47: Hence, we have a contradiction to the initial hypothesis. Necessarily therefore, there must be at least one prime within the interval c for all n ∈ N | n ≥ 47. Table 1 lists all values of n s.t. 11 ≤ n ≤ 53. Evidently, every possible sub-interval contains at least one prime number. Thus we deduce that Theorem 2.6 holds in this range as well. Consequently Theorem 2.6 holds as stated for all n ∈ N | n ≥ 11, thus completing the proof.

Remark 2.2.
We may now slightly relax the function Gs (n) by dropping the Floor function, if needed.
is valid for all p n ∈ N | p n ≥ 2. Where n ∈ N | n ≥ 1 is the index of the n-th prime number. Proof.
Now, (p n ) (n+1) = (p n ) n p n . Therefore, upon substitution into 3.2 we obtain: Since the prime gap g = p (n+1) − p n thus, p (n+1) = p n + g. Hence, from 3.3 we have: Taking logs, By the Theorem 2.6, maximal prime gaps are bounded above by the Supremum bound: Therefore, substituting SUP for the prime gaps g into 3.5, we obtain: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 June 2020 doi:10.20944/preprints202006.0366.v1 Let's examine in turn, the first term of 3.6. By the PNT we have that: (3.8) n log n + n (log log n − 1) < p n < n log n + n log log n ∀n ∈ N| n ≥ 6 Hence, from the Inequality 3.8, we have that: (3.9) p n < n log n + n log log n Clearly, both sides of the inequality 3.9 diverge as n → ∞ therefore, we substitute RHS of 3.9 into the first term of 3.6 thereby obtaining: (3.10) log p n n ≤ log (n log n + n log log n) n = log n + log (log n) + log n Now, for any n ∈ N | n ≥ 6, the limit of 3.10 by the L'Hôpital's rule is: Clearly, both n and log p n for all n ∈ N| n ≥ 1 are positive divergent functions. Consequently, due to the fact that: this of course implies n ≫ log p n for all n ∈ N | n ≥ 6. Table 2 demonstrates that n > log p n for all n ∈ N | 1 ≤ n ≤ 6. Consequently, n ≫ log p n for all n ∈ N | n ≥ 1. Now, Which implies that Inequality 3.6, (3.14) log p n n > log ≡ (n + 1) log p n > n log (p n + SUP) Suppose that the inequality 3.14 is false. First we exponentiate both sides of the inequality. Consequently, in accordance with the hypothesis for p n ≥ 11: However, at p 5 = 11 the difference attains ∼ 1.13722 × 10 6 and diverges exponentially. Since the difference of terms is positive, we apply the Cauchy's Root Test: Hence, the Cauchy's Root Test diverges, with the rate of divergence ∝ k p n | k ∼ 1. This implies that a series formulated from the terms of the difference 3.15 diverges. Consequently, we have a contradiction to the initial hypothesis. Hence, it implies that: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 June 2020 doi:10.20944/preprints202006.0366.v1 A straightforward computer calculation verifies that the relation 3.17 holds in the interval 2 ≤ p n ≤ 11. Now, since p (n+1) is defined as p (n+1) = p n + g, therefore, from the inequality 3.4, 3.6 and 3.17 we derive accordingly: The property of this function, is in a way analogous to the property of the  For every n ∈ N | n ≥ 11 the maximal prime gaps satisfy the inequality: where p (n) is the n-th prime number.

Proof.
The weak form of the Firoozbakht's Maximal Prime Gaps Bound, for all n ∈ N | n ≥ 11 asserts: The difference of the function FW n and the Supremum, defined by Theorem 2.6 is given by: The factors: This implies that for all p ∈ N | p ≥ 7 the weak Firoozbakht's maximal prime gaps bound FW n lies above the Supremum, vide Table 4 in the Appendix, Consequently by Theorem 2.6, the weak Firoozbakht's maximal prime gaps bound holds for all p ∈ N | p ≥ 11: For every n ∈ N | n ≥ 37 the maximal prime gaps satisfy the inequality: Proof.
The strong form of the Firoozbakht's Maximal Prime Gaps Bound, for all n ∈ N | n ≥ 37 asserts: The difference of FS n above and the Supremum defined by Theorem 2.6 is given by: The factors: (3.26) (log 10) 2 − 5 ∼ 0.301898 as well as 2 (log 10) − 15 8 ∼ 2.73017 Consequently, it is a matter of time before the leading quadratic term will begin to 'play the first violin'. The point where the difference 3.25 becomes positive, hence the point where the Supremum prime gaps bound intersects the stronger form of the Firoozbakht's prime gaps bound and remains below it, occurs: Consequently, by Theorem 2.6, the stronger form of the Firoozbakht's maximal prime gaps bound holds for all p ∈ N | p > 458 034 213. For all p ∈ N | 37 ≤ p ≤ 458 034 213, a computer calculation verifies that the stronger form of the Firoozbakht's maximal prime gaps bound holds in this range, vide Table 3 below and Table 4 in the Appendix. Therefore the stronger form of the Firoozbakht maximal prime gaps bound holds for all p ∈ N | p ≥ 37. Firoozbakht's Hypothesis is consistent with the Shank's Asymptotic Equality of Record Gaps Conjecture although it exposes a flaw and inconsistency in the Maier-Granville argument [14]: (3.27) g p k = p (k+1) − p k < M (log p k ) 2 with the limit as k tends to infinity: