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# A conjecture on consecutive primes – II

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Continuing with the previous post in this topic, a stronger form of conjecture on the lower bound of Corollary 1 is as follows

ConjectureLet $p_n$ be the n-th prime. The for $n \ge 32$,

$\displaystyle{p_n^{\frac{1}{n}} > \Big(1+\frac{1}{n^2}\Big) p_{n+1}^{\frac{1}{n+1}} }$.

The above conjecture implies that for all sufficiently large $n$,

$p_{n+1} - p_n < (\ln p_n - 1)(\ln p_n - \ln\ln n)$.

Prof. Marek Wolf, Institute of Theoretical Physics, Wroclaw, Poland has verified the above conjecture for primes up to $2^{44}$.