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Andrica’s Conjecture: Square-Root Gaps of Consecutive Primes

Prove that, for every n ≥ 1 where p_n denotes the nth prime number, the inequality sqrt(p_{n+1}) − sqrt(p_n) < 1 holds.

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Background

Andrica’s conjecture formulates a bound on the difference of square roots of consecutive primes, which implies strong constraints on prime gaps.

The paper cites Andrica’s conjecture among standard open problems and later presents corollaries on square-root differences derived from the claimed prime-gap bound.

References

Besides, Brocard conjectured that if $p_n$ is the $n$th prime with $n\ge2$, than there are at least four primes between $p_n2$ and $p_{n+1}2$; similarly, Andrica(1986) conjectured that if $p_n$ is the $n$th prime, then $\sqrt{p_{n+1}-\sqrt{p_n} < 1$.

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