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Firoozbakht’s Conjecture: Strict Inequality p_{n+1}^n < p_n^{n+1}

Show that, for every n ≥ 1 where p_n denotes the nth prime number, the strict inequality p_{n+1}^n < p_n^{n+1} holds.

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Background

Firoozbakht’s conjecture posits a strong monotonicity property equivalent to the sequence p_n{1/n} being strictly decreasing, implying very small prime gaps.

The paper references this conjecture alongside Cramér’s to highlight standard benchmarks for prime-gap size; the claimed upper bounds, if valid, would have implications for such inequalities.

References

Cram{e}r(1936) conjectured that given a prime $p_n$, then $\limsup_{n \to \infty} \frac{p_{n+1}-p_n}{\log2{p_n}=1$(); on the other hand, Firoozbakht(1982) conjectured that if $p_n$ is the $n$th prime, then $p_{n+1}n < p_n{n+1}$, both of which imply a prime gap stronger than the previous ones.

— On the Maximal Gap between Primes (2510.17065 - Wang, 20 Oct 2025) in Section 1 (Introduction)