# Strong version of Andrica's conjecture

## TL;DR

This paper verifies a strong version of Andrica's conjecture for all primes below 2^64, and shows its relation to other prime conjectures, providing unconditional and explicit validation up to very large bounds.

## Contribution

The paper unconditionally verifies a strong form of Andrica's conjecture and related conjectures for all primes below 2^64, establishing explicit bounds and connections among these conjectures.

## Key findings

- Verification of the strong Andrica's conjecture for primes below 2^64.
- Relation of the strong Andrica's conjecture to Oppermann's, Legendre's, and Brocard's conjectures.
- Explicit bounds where these conjectures are verified unconditionally.

## 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 $p <2^{32} \approx 4.294 \times 10^9$.