Farhadian's conjecture

In number theory, the Farhadian's conjecture (or Reza's conjecture) is a conjecture about the consecutive primes that presented by Reza Farhadian in 2016.

Let $p_{1}=2,p_{2}=3,p_{3}=5,p_{4}=7,\ldots$ be the successive prime numbers in their natural ordering, and in general, let $p_{n}$ be the $n$-th prime. Farhadian's conjecture states that for every $n>4$, always we have:

$p_{n}^{(p_{n+1}/p_{n})^{n}}<n^{p_{n}}$,

i. e, for $n>4$, the $p_{n+1}$ bounded by the following conjectural inequality:

$p_{n+1}<p_{n}\left(p_{n}{\frac {\log n}{\log p_{n}}}\right)^{1/n}.$

There are many other conjectures on consecutive primes $p_{n}$ and $p_{n+1}$, with the best known of these being Cramér's $p_{n+1}<p_{n}+O(\log ^{2}p_{n})$, Firoozbakht’s $p_{n+1}<p_{n}p_{n}^{1/n}$ and Granville's $p_{n+1}<p_{n}+M\log ^{2}p_{n}$ (for some $M>1$ ) conjectures. We know that the Firoozbakht's conjecture is stronger than Cramér's conjecture and also Cramér's conjecture is Stronger than Granville's conjecture. On the other hand, the Farhadian's conjecture is stronger than Firoozbakht's conjecture; Because $\left(p_{n}{\frac {\log n}{\log p_{n}}}\right)^{1/n}<p_{n}^{1/n}$. Thus, the Farhadian's conjecture is stronger than Cramér's and Granville's conjectures.

References

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