2020/11/30 by Bernard Montaron, Montaron, Bernard
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2011.14653
openalex publication_date 2020/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Infinite exponential sequences of distinct prime numbers of the form \lfloor a cnd+b\rfloor, n≥ 0, are proved to exist for well chosen real constants a>0, b, c>1, d>1, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with a=1, b ∈ \0,1\, that have the smallest possible constant c given d>1, and sequences with the smallest possible d, given c=2. In particular, we prove the existence of the four infinite exponential prime sequences u0(n)=\lfloor c0n√(n)\rfloor, n≥ 1, with c0=2.0073340803..., u1(n)=1+\lfloor c1n√(n)\rfloor, n≥ 0, with c1=2.2679962677..., v0(n)=\lfloor 2^nd0\rfloor, n≥ 1, with d0=1.5039285240..., and v1(n)=1+\lfloor 2^nd1\rfloor, n≥ 0, with d1=1.7355149500....