The Nicholson's conjecture

This research paper discusses the distribution of prime numbers, from the point of view of the Nicholson's conjecture of 2013. The proof of the conjecture, permits to develop and establish a Supremum bound, on the difference of terms of the conjecture. Nicholson's conjecture belongs to the class of the strongest bounds on maximal prime gaps.


Abstract.
This research paper discusses the distribution of the prime numbers, from the point of view of the Nicholson's Conjecture of 2013: ( The proof of the conjecture permits to develop and establish a Supremum bound on: ) n

≤ SU P
Nicholson's Conjecture belongs to the class of the strongest bounds on maximal prime gaps.
Key words and phrases. Distribution of primes, maximal prime gaps upper bound, Nicholson's conjecture, Prime Number Theorem.

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 n , p (n+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: A variant of logarithmic scaling of the horizontal axis is given by: Definition 1.2 (Scaling factor). ξ = log 10 ( n 24 ) log 10 (24) Figure 1. shows the graphs of Gs (n) (red) and the actual maximal gaps (black) with respect to ξ as given by Def. 1.2. The graph has been produced on the basis of data obtained from C. Caldwell as well as from T. Nicely tables of maximal prime gaps. The occurrence of the gap 64; g = 1 131, beginning at the prime 1 693 182 318 746 371, visible on the graph at ξ ≈ 10.03 exhibits the difference Gs (n) − g = 0.0132662.

Theorem 1.3 (Maximal Prime Gaps Supremum and Infimum for primes).
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 upper bound UB is given by:

The Nicholson's Conjecture
John Nicholson [17] in 2013 postulated a conjecture concerning prime numbers distribution and the prime gaps upper bound. His conjecture represents a moderate improvement, as compared to the weak form of the Firoozbakht's conjecture of 1982, please refer to Feliksiak [10].
The Nicholson's conjecture is valid for all p n ∈ N | p n ≥ 11: Proof. Inequality 2.1 is equivalent to: Hence, we obtain the expression for an upper bound on an arbitrary prime gap: It is obvious, that Nicholson's conjecture worst case scenario, occurs at the Maximal Prime Gaps. If Theorem 2.1 holds at the Maximal Prime Gaps, necessarily, it must hold at every other prime gap. Suppose that the inequality 2.3 is false for p n ∈ N | p n ≥ 1 657. This implies that: Theorem 1.3, specifies the upper bound on maximal prime gaps, therefore, in accordance with the hypothesis, we must have: Cauchy's Root test at p n = 1 657: ) attains approx. 1.0008404394611654 and asymptotically, strictly from above, tends to 1, as p n increases unboundedly. Please refer to Figure 2a. By the definition of the Cauchy's Root Test therefore, this implies that the sequence a n diverges. Consequently, since in accordance with the hypothesis: < 0 the value of inequality 2.7 must decrease as p n increases unboundedly. However, at p n = 1 657 inequality 2.7 attains approx. 1.24411. Since it is a divergent sequence, the difference 2.7 continues to increase as p n increases unboundedly. Please refer to Figure 2b. Therefore, we have a contradiction to the initial hypothesis. Hence, necessarily for all p n ∈ N | p n ≥ 1 657 the inequality is valid: which implies: For the range of p n ∈ N | 11 ≤ p n ≤ 1 657, direct computer evaluation evidently confirms, that inequality 2.9 holds in this range as well. Please refer to Figure 3. Hence, Theorem 2.1 holds for all p n ∈ N | p n ≥ 11. Concluding the proof.  . The drawing shows the graph of the difference of terms of the inequality 2.9, drawn over p n ∈ N | 11 ≤ p n ≤ 1 987.

Theorem 2.2 (Nicholson's Conjecture -Supremum Bound).
The Supremum, on the difference of terms of the Nicholson's Conjecture, for all p n ∈ N | p n ≥ 2 is given by: and γ is the Euler-Mascheroni constant.

Proof.
Suppose that for p n ≥ 2 Theorem 2.2 is false. This implies that: However, inequality 2.11 at p n = 2 attains approx. 1.90643 and asymptotically, strictly from above tends towards 0 as p n increases unboundedly. Please refer to Figure 4a. The Cauchy's Root Test of inequality 2.11, converges asymptotically, strictly from below to 1. Hence the test is inconclusive. Therefore, to determine the case for convergence/divergence, we implement: At p n = 2 the difference 2.12 attains approx. −1.47546 and rapidly diverges at a rate ∝ k p n with k ∼ 1, for p n ∈ N | p n ≥ 2. Please refer to Figure 4b. This implies that: Hence, the L.H. of 2.13 tends to zero slower than the R.H. i.e. 1/p n . The divergence of the sum: (2.14) had already been proven by Euler. This implies, that inequality 2.11, converges asymptotically, strictly from above to zero, as p n tends to infinity. Consequently, we have a contradiction to the initial hypothesis. Therefore, for all p n ≥ 2 Theorem 2.2 holds as stated, concluding the proof.

Corollary 2.3 (Upper bound on the function log n).
From Theorem 2.2, the Upper bound on log n for all n ∈ N | n ≥ 1 is given by: