Strong version of Andrica's conjecture
Matt Visser
TL;DR
The paper formulates a strong version of Andrica's conjecture and situates it within a network of related conjectures (Oppermann, Legendre, Brocard). It demonstrates an unconditional, explicit verification of the strong Andrica conjecture for all primes below the 81st maximal prime gap, i.e. $p<2^{64}$, and shows that this implies Oppermann and the strong/standard Legendre and Brocard conjectures in the same range, with Brocard verifications valid up to $p<2^{32}$. By chaining implications among these conjectures, the work yields broad, verifiable regions for several prime-gap statements and highlights how future maximal-gap data will extend these regions.
Abstract
A strong version of Andrica's conjecture can be formulated as follows: Except for $p_n\in\{3,7,13,23,31,113\}$, that is $n\in\{2,4,6,9,11,30\}$, one has$\sqrt{p_{n+1}}-\sqrt{p_n} < \frac{1}{2}.$ While a proof is far out of reach I shall show that this strong version of Andrica's conjecture is unconditionally and explicitly verified for all primes below the location of the 81$^{st}$ maximal prime gap, certainly for all primes $p <2^{64}\approx 1.844\times 10^{19}$. Furthermore this strong Andrica conjecture is slightly stronger than Oppermann's conjecture --- which in turn is slightly stronger than both the strong and standard Legendre conjectures, and the strong and standard Brocard conjectures. Thus the Oppermann conjecture, and strong and standard Legendre conjectures, are all unconditionally and explicitly verified for all primes $p <2^{64}\approx1.844\times 10^{19}$. Similarly, the strong and standard Brocard conjectures are unconditionally and explicitly verified for all primes $p <2^{32} \approx 4.294 \times 10^9$.