On the maximal gap between consecutive primes: an 𝑂 (ln 𝑝) bound via additive basis properties of K-indices with 4𝑘 ± 1 prime

It has been proved by the author earlier, that the generating set of K-indices of primes K = {𝑘 ∈ N | 4𝑘 − 1 ∈ P or 4𝑘 + 1 ∈ P} is a complete additive basis of order 2. From this property, we derive an upper bound for the maximal gap between consecutive primes: 𝑔𝑃(𝑝𝑛) = 𝑝𝑛+1 − 𝑝𝑛 = 𝑂(ln 𝑝𝑛). In particular, every sufficiently large gap satisfies 𝑝𝑛+1 − 𝑝𝑛 ≤ 𝐶 ln 𝑝𝑛 with an absolute constant 𝐶 > 0. Idea of the proof. Assuming, for contradiction, that a gap 𝑘𝑖+1 − 𝑘𝑖 in K exceeds 𝐶 ln 𝑘𝑖 , we consider the number 𝑁 = 2𝑘𝑖+1. Using the fact that K is an additive basis, 𝑁 must be representable as 𝑎 + 𝑏 with 𝑎, 𝑏 ∈ K. A case-by-case analysis of the possible positions of 𝑎 and 𝑏 relative to 𝑘𝑖 and 𝑘𝑖+1 shows that such a representation is impossible. Hence, all gaps in K are 𝑂(ln 𝑘𝑖), and via the form 𝑝 = 4𝑘 ± 1 we obtain the same bound for prime gaps. Consequences. This estimate immediately implies several classical conjectures for all sufficiently large indices: Legendre’s conjecture (a prime between 𝑛2 and (𝑛 + 1)2 ), Brocard’s conjecture (a prime between 𝑝𝑛2 and 𝑝𝑛+12 ), and Opperman’s conjecture (primes in [𝑛(𝑛 − 1), 𝑛2 ] and [𝑛2 , 𝑛(𝑛 + 1)]). The bound also improves unconditionally upon the best known analytic results (Baker–Harman–Pintz, 𝑂(𝑝0.525)) and even upon the Cramer conjecture (𝑂(ln2𝑝)). The proof is elementary; use no analytic number theory; it relies only on the additive basis property of K and a combinatorial analysis of gaps.