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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 $n$,

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

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