On the composites among [ξ7ⁿ]

Abstract

We prove that for every real $\xi>0$ the sequence of integer parts $[\xi 7^{n}]$, $n=0,1,2,\dots$, contains infinitely many composite numbers. This is the case $a=7$ of a problem going back to Forman and Shapiro (1967), stated by Dubickas and Novikas to be open for every integer $a\ge 7$. By the same certificate, applied to prime chains directly, there is no infinite right truncatable prime in base~$7$. The proof reduces the problem to a single finite computation: modulo $M=\prod_{p\le 31}p$, the digraph of residues traversed by a sequence avoiding all prime factors of $M$ has a core in which no strongly connected component contains two distinct cycles, so every avoiding digit word is ultimately periodic and an elementary return theorem of Dubickas produces composite terms. The set $\{p\le 31\}$ is \emph{not} an unavoidable set of divisors --- the method succeeds although, for this base, no unavoidable set is expected to exist. All results, including the finite computation, are verified in Lean~4; the computation is checked by three quarantined \texttt{native\_decide} evaluations, and everything else depends only on Lean's three standard axioms.

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