Cramér\'s conjecture

Cramér's conjecture by Thomas McClure I

Introduction

This paper is on Cramér's conjecture.

II

Cramér's conjecture

https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936,[1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that

p_{n+1}-p_n=O((\log p_n)^2),\ where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement

\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1, and this formulation is often called Cramér's conjecture in the literature.

Neither form of Cramér's conjecture has yet been proven or disproven.

Cramér gave a conditional proof of the much weaker statement that

p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n) on the assumption of the Riemann hypothesis.[1]

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,[2] lim{n,%} = {p_{n+1}-p_n}/{log p_n} = 1/log 1 \limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.

lim{n,%} = {p_{n+1}-p_n}/{log p_n} * {ln p_n/ln p_n} = 1/log 1 lim{n,%} = {p_{n+1}-p_n}/{ln p_n} * {ln p_n/log p_n} = 1/log 1 lim{n,%} = {p_{n+1}-p_n}/{ln p_n} = {log p_n/ln p_n} * 1/log 1 lim{n,%} = {p_{n+1}-p_n}/{ln p_n} = {log (p_n)/1} / ln p_n lim{n,%} = {p_{n+1}-p_n} = log p_n log p_n = ln p_n/ln 10 == n/pi(n) * 1/ln 10 [1.5]^2

1/[2.3]

= 1/

Hence, the limit of the prime gap is equal to the log of p_n , which is a ratio of ln p_n/ln 10 by the change of base formula. https://en.wikipedia.org/wiki/Logarithm#Change_of_base The inverse ln 10 is the inverse square of 1.5, so the log p_n is about (1/1.5*1.5) ln p_n. That is, log p_n *1.5 == ln p_n/1.5 .

III

Conclusion

This paper is on Cramér's conjecture. By deduction lim{n,%} = {p_{n+1}-p_n} = log p_n lim{n,%} = {p_{n+1}-p_n} = ln p_n/ln 10 This formula is much simpler than other formulae for the conjecture.