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A conjecture on consecutive primes

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In 1982, Farideh Firoozbakht made an interesting conjecture that the sequence p_n ^{1/n} decreases i.e. for all n \ge 1,

p_n ^{\frac{1}{ n}} > p_{n+1} ^{\frac{1}{n+1}}.

I present a stronger form of Firoozbakht’s conjecture.

Conjecture: Let p_n be the n-th prime and let r(n) = p_n^{1/n}.

\displaystyle{\lim_{n \rightarrow \infty} \frac{n^2}{\ln n} \Big(\frac{r(n)}{r(n+1)}-1\Big) = 1}.

If the above conjecture is true than we can have explicit bounds on p_n ^{1/n} in terms of p_{n+1}.

Corollary 1: For every \epsilon, 0<\epsilon<1, there exists a sufficiently large natural number N_{\epsilon}, which depends only on \epsilon, such that for all n>N_{\epsilon},

(1+n^{-2}) p_{n+1} ^{\frac{1}{n+1}} < p_n ^{\frac{1}{n}} < (1+n^{-2+\epsilon})p_{n+1} ^{\frac{1}{n+1}}.

The above inequality would imply Cramer’s conjecture and in fact we have a stronger bound on the gap between consecutive primes.

Corollary 2: For all sufficient large n,

p_{n+1} - p_n < \ln ^2 p_n - 2\ln p_n + 1.

References

[1] http://arxiv.org/abs/1010.1399

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