Strong version of Andrica's conjecture

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 $p<2^{64}\approx 1.844\times 10^{19}$. Similarly, the strong and standard Brocard conjectures are unconditionally and explicitly verified for all primes $p<2^{32}\approx 4.294\times 10^9$.

V1: 8 pages. Sequences: A005250 A002386 A005669 A000101 A107578 A050216 A050216; V2: Now 11 pages; some discussion and references added. V3: Still 11 pages. Updated to reflect best currently known bound on the location of the 81st maximal prime gap (somewhere above 2^{64})

References in corpus (3)

• Estimates of Some Functions Over Primes without R.H.
• Variants on Andrica's conjecture with and without the Riemann hypothesis
• A Short Note on Gaps between Powers of Consecutive Primes

Cited by in corpus (1)

• Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap