Prime Gap Conjecture

Prime Gap Conjectures by Thomas McClure I

Introduction

This paper is written on Prime Gap Conjectures.

II

Conjectures about gaps between primes

https://en.wikipedia.org/wiki/Prime_gap Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap gn satisfies

g_n = O(\sqrt{p_n} \ln p_n), using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that

g_n = O\left((\ln p_n)^2\right). At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

Firoozbakht's conjecture states that p_{n}^{1/n}\, (where p_n\, is the nth prime) is a strictly decreasing function of n, i.e.,

p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1. If this conjecture is true, then the function g_n = p_{n+1} - p_n satisfies g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4. [24]

prime gap g_n = {p_(n+1) - p_n} {p_(n+1) - p_n} < p_{n+1} < p_n^{1 + 1/n} p_n < p_n^{1 + 1/n} ln (p_n) < {1 + 1/n} (ln (p_n) 1 < {1 + 1/n} The above inequality is a restatement of Firoozbakht’s conjecture.

This is one of the strongest upper bound ever conjectured for prime gaps. Moreover, this conjecture implies Cramér's conjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of record gaps.[25]

By using tables of maximal gaps, Firoozbakht's conjecture has been verified for all primes below 4×1018.[26]

Mean while, the Oppermann's conjecture is a conjecture which is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is

g_n < \sqrt{p_n}\, . Andrica's conjecture, which is a weaker conjecture to Oppermann's, states that[20]

g_n < 2\sqrt{p_n} + 1.\, This is a slight strengthening of Legendre's conjecture that between

successive square numbers there is always a prime. [p_{n+1} < 2p_n]

[Bertrand's postulate]

[p_{n+1} - p_n < p_n] [g_n < p_n] [g_n < root{p_n} < 2*root{p_n} + 1] [g_n < root{p_n}[2 + root{p_n}/2] [g_n < root{p_n} < [2 + root{p_n}/2}] [g_n < root{p_n} < p_n] III

Conclusion

This paper is written on Prime Gap Conjectures. For prime gap g_n ; [g_n < p_n] [g_n < root{p_n} < p_n]