Primes in LCM recurrences

Abstract

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a strengthening of Linnik's theorem on the least prime in an arithmetic progression that lies beyond current reach. We develop a Companion--Sieve framework that reduces the conjecture to an equidistribution problem for primes in the progression $-1\mathrm{mod}p$, and applying the Bombieri--Vinogradov theorem we prove unconditionally that the conjecture holds for a set of integers of asymptotic density $1$. We also give an effective finite reduction showing that any counterexample beyond a computable threshold involves only large prime factors. A closely related recurrence turns out to encode twin prime pairs through its increment pattern, and we prove a conditional density-$1$ result for it under a prime-index detection hypothesis, using an upper-bound Selberg sieve estimate for twin primes in arithmetic progressions. The analysis also leads to three new conjectures on the distribution of primes in arithmetic progressions.

v3, 25 pages. Minor changes and use of Selberg sieve