Intervals without primes near an iterated linear recurrence sequence
Kota Saito
TL;DR
This work studies intervals around iterated inhomogeneous linear recurrence sequences (ILRS) and proves that, under explicit reversibility and growth conditions on the composing sequences $R_j(n)$, the intervals $\big(|f(n)|-c\log n, |f(n)|+c\log n\big)$ contain no primes for infinitely many $n$, where $f(n)=(R_0\;∘\;⋯\;∘\;R_M)(n)$ and $c$ depends only on the recurrence orders. The authors establish modular-periodicity lemmas for ILRS modulo primes, extend them to iterated compositions, and leverage a PNT-type estimate to force divisibility by carefully chosen primes along arithmetic progressions, yielding infinite prime-void windows with a quantified error term. A corollary shows that for any Pisot or Salem number $α$, the numbers $\lfloor α^{(R_1\;∘\;⋯\;∘\;R_M)(n)} \rfloor$ are composite for infinitely many $n$, by expressing the floor as a sum of an ILRS component and a bounded trace term. The results also recover and extend earlier work (D1, D2) in the base case $M=0$ and provide explicit constants for the size of the prime-free intervals.
Abstract
Let $M$ be a fixed positive integer. Let $(R_{j}(n))_{n\ge 1}$ be a linear recurrence sequence for every $j=0,1,\ldots, M$, and we set $f(n)=(R_0\circ \cdots \circ R_M)(n)$, where $(S\circ T)(n)= S(T(n))$. In this paper, we obtain sufficient conditions on $(R_{0}(n))_{n\ge 1},\ldots, (R_{M}(n))_{n\ge 1}$ so that the intervals $(|f(n)|-c\log n, |f(n)|+c\log n)$ do not contain any prime numbers for infinitely many integers $n\ge 1$, where $c$ is an explicit positive constant depending only on the orders of $R_0,\ldots, R_M$. As a corollary, we show that if for each $j=1,2,\ldots, M$, the sequence $(R_j(n))_{n\ge 1}$ is positive, strictly increasing, and the constant term of its characteristic polynomial is $\pm 1$, then for every Pisot or Salem number $α$, the numbers $\lfloor α^{(R_1\circ \cdots \circ R_M)(n)} \rfloor $ are composite for infinitely many integers $n\ge 1$.